This is sort of an off the cuff ramble of an answer for a topic which deserves more careful thinking, so I might make some hand-wavy statements and grand sweeping claims which I will not endorse later, but:
First off, I feel that it's a little unhelpful to frame the question this way. It implicitly forces answers to conflate some fairly separate concepts together 1) The System 2) leftists 3) critiques of EA.
Here's a similarly sort of unhelpful way to ask a question:
What are these "cognitive biases" that effective altruist critiques of veganism are seeking to make us aware of?
How would you answer?
Most effective altruists support veganism! The central insight motivating most vegan practices is similar to the central insight of EA. Don't lose sight of that just because some branches of effective altruists think AI risk rather than veganism is a best possible way to go about doing good, and cite cognitive biases as the reason why people might not realize that AI risk is the top priority.
Cognitive Biases are a highly useful but fully generalizable concept that can be used to support or critique literally anything. You should seek to understand cognitive biases them in their own right...not only in the light of how someone has used them to form a "critique of veganism" by advocating for AI risk instead.
That's how you'd answer, right? So, in answer to your question:
What exactly is the system EA's (leftist) critics are seeking to change?
Most ideologically consistent leftists support EA, or would begin supporting it once they learn what it is. Utilitarianism / widening of the moral circle is very similar to ordinary lefty egalitarianism. Don't lose sight of that just because some branches of the left don't think some particular EA method are the best possible way to save the world, and cite Failure to Challenge the System as the reason.
The System is a highly useful but fully generalizable concept that can be used to support or critique literally anything. You should seek to understand it in its own right...not only in the light of how someone might invoke it to form a "critique of (non-systemic) effective altruism" by advocating for systemic change instead
I hope this analogy made my point - this question implicitly exaggerates a very minor conflict, setting up an oppositional framework which does not really need to exist.
...okay, so to actually attempt to answer the quesiton rather than subvert it. Please note that the following are not my own views, but a fairly off the cuff representation of my understanding of a set of views that other people hold. Some of these are "oversimplified" versions of views that I do roughly hold, while others are views that I think are false or misguided.
What is the system?: Here's one oversimplified version of the story: from the lower to upper paleolithic, egalitarian hunter gatherers gradually depleted the natural ecology. Prior to the depletion, generally most able bodied persons could easily provide for themselves and several dependents via foraging. Therefore, it was difficult for anyone to coerce anyone else, no concepts of private property were developed, and people weren't too fussy about who was related to whom.
In the neolithic, the ecology was generally getting depleted and resources were getting scarce. Hard work and farming became increasingly necessary to survive and people had incentive to violently hoard land, hoard resources, and control the labor of others. "The System" is the power structures that emerged thereby. It includes concepts of private property, slavery, marriage (which was generally a form of slavery), social control of reproduction, social control of sex, caste, class, racism, etc - all mechanisms ultimately meant to justify the power held by the powerful. Much like cognitive biases, these ideas are deeply built into the way all of us think, and distort our judgement. (E.g. do you believe "stealing" is wrong? Some might argue that this is the cultural programming of The System talking. Without conceptions of property, there can be no notion of stealing)
Despite resource scarcity declining due to tech advance, the bulk of human societies are still operating off those neolithic power hierarchies, and the attending harmful structures and concepts are still in place. "Changing the system" often implies steps to re-equalizing the distribution of power and resources, or otherwise dismantling the structures that keep power in the hands of the powerful.
By insisting that the circle of moral concern includes all of humanity (at least), and actively engaging in a process which redistributes resources to the global poor, effective altruists would generally be considered as a source of positive contributing to the dismantling of "The System". I do think the average leftist would think Effective Altruism, properly pitched, is generally a good idea - As would the average person regardless of ideology, realistically, if you stuck to the basic premises and didn't get too into some of the more unusual conclusions that they sometimes are taken to.
So how come some common left critiques of EAs invoke "The System"?:
Again, I don't (entirely) agree with all these views, I'm explaining them.
1) Back when the public perception of EA was that it was about "earning to give" and "donating"...especially when it seemed like "earning to give" meant directing your talent to extractive corporate institutions, the critique was that donations do not actually alter the system of power. Consider that a feudal lord may "give" alms to the serf out of noblesse oblige, but the fundamentally extractive relationship between the lord and serf remains unchanged. I put "give" in quotes because, if you really want to understand The System, you have to stop implicitly thinking of the "lord's" "ownership" of the things they "nobly" "give" to the "serf" as in any way legitimate in the first place. The lord and serf may both conceptualize this exchange as the lord showing kindness towards the serf, but the reality is that the lord, or his ancestors, actually create and perpetuate the situation in the first place. Imagine the circularity of the lord calculating he had made a magnanimous "impact" by giving the serf a bit of the gold... that was won by trading the grain which the serf had toiled for in the first place. Earning to give is a little reminiscent of this...particularly in fields like finance, where you're essentially working for the "lord" in this analogy.
2) Corporate environments maximize profit. Effective altruists maximize impact. As both these things are ultimately geared towards maximizing something that ultimately boils down to a number, effective altruist language often sounds an awful lot like corporate language, and people who "succeed" in effective altruism look and sound an awful lot like people who "succeed" in corporate environments. This breeds a sense of distrust. There's a long history within leftism of groups of people "selling out" - claiming to try to change the system from inside, but then turning their backs on the powerless once they got power. To some degree, this similarity may create distasteful perceptions of a person's "value" within effective altruism that is analogous to the distasteful perception of a person's "value" in a capitalist society. (E.g. capitalist society treats people who are good at earning money as sort of morally superior. Changing "earning money" to "causing impact" can cause similarly wrong thinking)
3) EAs to some extent come off as viewing the global poor as "people to help" rather than "people to empower". The effective altruist themself is viewed as the hero and agent of change, not the people they are helping. There is not that much discussion of the people we are helping as agents of change who might play an important part in their own liberation. (This last one happens to be a critique I personally agree with fairly wholeheartedly, and plan to write more on later)
To the extent the systemic change criticism of EA is incorrect, as EA enters the policy arena more and more, we will once again come in friction with leftist (and other political movements), unlike EA has since its inception. The difference this time is we would be asserting the systemic change we're pursuing is more effective (and/or in other ways better) than the systemic change other movements are engaging in. And if that's the case, I think EA needs to engage the communities of our critics just as critically as they have engaged us. This is something I've begun working on myself.
I would strongly recommend not creating a false dichotomy between "EA" and "Leftists", and setting up these things as somehow opposed or at odds. I'm approximately an EA. I'm approximately a leftist. While there are leftist-style critiques of EA, and EA-style critiques of leftism, I wouldn't say that there's any particular tension between these frameworks.
There is really no need to draw lines and label things according to ideology in that manner. I think the most productive reply to a "X-ist" critique of EA is an X-ist support of EA, or better yet, a re-purposing of EA to fulfill X-ist values. (Yes, there are some value systems for which this cannot work...but the egalitarian left is definitely not among those value systems)
to the extent the systemic change criticism of EA is correct, EA should internalize this criticism, and should effectively change socioeconomic systems better than leftists ever expected from us
And to that I would add, don't needlessly frame EA as fundamentally in opposition to anyone's values. EA can be framework for figuring out strategic ways to fulfill your values regardless of what those values are. (Up to a point - but again, "leftists" are well within the pale of that point.)
...and perhaps better than leftist political movements themselves (lots of them don't appear to be active or at least effective in actually changing "the system" they themselves criticize EA for neglecting).
Well, I think this is an unhelpful tone. It is, again, setting up EA as something different and better than leftism, rather than a way for us to fulfill our values - even if our values aren't all exactly the same as each others. This isn't particular to leftism. If you wanted the members of a church congregation to donate to Givewell, you should focus on shared values of charity, not "EAs could save more souls than Christianity ever could". The goal for EA is not to engage against other ideologies, the goal (to the extent that EA ideas are good and true, which obviously they may not all be) is to become part of the fabric of common sense by which other ideologies operate and try to perpetuate their goals.
Beyond the tone it's also just not true, in my opinion. Seems to me that social change does in fact occur constantly due to political movements, all the time. What's more, I'm pretty sure that the widespread acceptance of the basic building block concepts of effective altruism (such as, all people are equally important) are largely due to these leftist social movements. I don't think it's a stretch to say that EA itself is at least in part among the products of these social movements.