I don’t think you have to been a total utilitarian to believe in “saving lives”. That is because an increase in population is not the main benefit of saving lives, and indeed many interventions that we think of as “saving lives” may have no or even a negative impact on total population.

Population dynamics are complicated and not that well understood, but one thing we know is that reduced infant and child mortality has a negative effect on birth rates. Intuitively you could think of this is a de-risking effect; if a parent wants to ensure that at least one or two kids make it to adulthood, then many more births are required under a high mortality environment.

This is an important and valuable question, thank you for raising it. I'll split my observations into two effects:

1. Malthusian effects
2. Benefits of scale

Malthusian effects

Other responses have referred to Malthusian effects, by which I mean the concern that with only finite resources, the resources will be spread between more people, and each person will have a worse quality of life.

Benefits of scale

Creating another person doesn't only create another mouth to feed. It also creates another source of ideas and creativity. 

For example, each new birth has the potential to become another Norman Borlaug (who is claimed to have saved a billion lives through his research). 

Even if 999,999 new people fail to come up with a ground-breaking innovation which makes the world better, if the millionth person does, it could allow everyone to benefit.

Of course, the flipside is that any new person could be the next Hitler/Stalin/insert-your-favourite-bad-guy-here.

Are the Norman Borlaugs winning over the Hitlers?

If you believe the data that seem to suggest that the world has been getting better over recent centuries, the answer seems to be yes.

There's also benefits around the fact that niche interests/needs are better accommodated at scale. If 0.01% of the population has a rare disease, and the population is 10 billion people, that's a million sufferers -- enough scale to incentivise scientific research. And if successful, maybe everyone is cured/treated. For a significantly smaller population that disease may remain untreated for a very long time.

Do the benefits of scale win over the Malthusian effects?

I don't think this is obvious, but I'm inclined to think the benefits of scale win. 

If we look at recent examples of challenges that humanity has faced, human ingenuity has managed a few good successes (the aforementioned example of Norman Borlaug and dwarf wheat; the cost effectiveness of solar power has improved dramatically in recent years; smallpox eradication; saving the ozone layer). Don't get me wrong, we still have more to do! But that suggests we want more brains, not less.

Furthermore, decisions we make today should be based on how the benefits of scale will work in the future, not how they were in the past. Will we be better able to use our ingenuity to solve big problems in the future? Some would argue that AI will make us better able to explore creative new solutions (not that everyone will agree on this).

Lastly, and this isn't really answering your question, but rather picking up on a comment of yours. You said that the idea that saving lives makes the world better is a "core assumption of the effective altruism movement". I don't think this is correct. EA is a movement built around using evidence and reason to do good. If the evidence showed that saving lives was bad, the essence of EA would be unchanged. Furthermore, lots of the practice would be unchanged too -- a lot of EA activity is not linked to saving lives.

I think a non-paradoxical answer is: "a world with 10 billion people is better, but probably not twice as good".

The reason it might not be quite twice as good is because of the resource issues you mention, that might decrease average welfare.

What limits population, is farmable land. You shouldn't worry about the rest. Malthusianism has been disproven many times. 

A world with 20 billion people is as good. Humanity will leave its cradle one day, and go to the stars. 

A world with 1 billion of people would be less innovative, sicker. If you look at past times: low population has not guaranteed anything to be better. 

Economic theory predicts more people actually will stimulate demand for services, innovation and more. 

As always population mostly has been limited by human organisation, technology, and food. Not by else. A 20 billion world is wholly liveable, more innovative, with slightly taller skyscrappers. 

You people need to learn more about economics. Humans are very creative at increasing life quality. My nearby shops now has a system locker to rent an electronic device 1 day that's automated, that's what economies of scale are.