There are five layers of logical error in your argument here, any one of which would invalidate your argument on its own:

1. In your examples with the US government and Amazon, the US government and Amazon could easily do this "middleman" verification work themselves. No middleman required.
2. Even if you use a middleman, the US government and Amazon totally control this situation - they own and completely control everything about these fictional currencies! - and so they are able to renege on their guarantees or otherwise cheat you at any time. A middleman can't stop them from doing that - only the law can, which is not reliant on crypto. So letting them manage the transactions, as suggested in point 1, isn't handing them any more power.
3. Even if you had to use a middleman, it wouldn't be much less expensive to leave all that verification to the government, and middleman fees are small anyway. You're zeroing in on a hypothetical problem that basically does not exist in reality! It's extremely rare to hear about PayPal or Visa or Mastercard defrauding their users, and these businesses are in competition to provide the lowest fees. Meanwhile, the crypto ecosystem is burdened by gas fees and massive electrical/computer hardware losses, depending on whether you use PoS or PoW. Plus, the legal infrastructure which prevents Visa from defrauding users doesn't exist yet for crypto, and if it did, then the arguments about crypto letting you evade regulation (i.e. commit money laundering) would disappear. There's no way to win here. Crypto is just worse, and it always will be.
4. You're arguing that crypto might be more trustworthy than these established middlemen, who have long histories of abiding by financial regulations? We all know that the crypto ecosystem is teeming with scams; it's a safe bet that scams would pop up to take advantage of these new cryptocurrencies.
5. Crypto technology is inefficient garbage. Here, a computer science professor explains how you can make a system that does everything Bitcoin does, with a tiny fraction of the electrical consumption, using a network of ten Raspberry Pi mini-computers: https://www.currentaffairs.org/2022/05/why-this-computer-scientist-says-all-cryptocurrency-should-die-in-a-fire Even if all my previous four arguments are nonsense, the US government or Amazon could save money and resources by using this extremely primitive system instead of crypto, and still get all the supposed benefits.

OK. Without evaluating your proposal for its pros and cons... there is no reason why the thing you propose should be implemented as a cryptocurrency. 

You're describing something which is in no way dependent on or enabled by blockchain tech.

Cryptocurrency is all bad.

Cryptocurrencies immediately self-organize into Ponzi schemes from the moment they are created, because when you create a speculative asset with no value other than speculative value, a Ponzi scheme is all that it can become.

The necessary incentive structure exists right from the start: people will want to buy low and sell high, but the asset's value is entirely imaginary (unlike that of land or gold, for example). So they speculate until the bubble pops and everyone tries to cash out in panic, and because the asset's "value" was literally 100% bubble-hype and 0% tangible economic value... well, that's a straightforward description of every Ponzi scheme ever.

I got in a discussion about this with another forum user here; rather than rehashing any more of what I've already said, I invite anyone reading this to check out the exchange as it happened: https://forum.effectivealtruism.org/posts/dQuaxGRkvEDXQLSNm/?commentId=ir9ihgeBkx4NTAvGD

Bitcoin is NOT like gold. Gold, unlike cryptocurrencies, is a tangible item that can be used for things. (Electronics, jewelry, etc.)

And unlike bitcoin-hoarding, with its massive losses in the forms of electricity and computer hardware, gold-hoarding isn't a massively negative-sum game. But that's beside the point: crypto is a Ponzi scheme and gold isn't because gold can do economic work and cryptocurrency can't.

I've been over Ethereum's fundamental toxicity - both the fundamental toxicity of PoS and the fundamental toxicity of cryptocurrencies in general - in my previous comments.

"How is this any different from buying stocks, or buying anything else for that matter?"

See point 3 in my previous comment.

"Crypto (or at least Ethereum) is part ownership of a technology."

No, it's not. You don't have part ownership of anything. You haven't bought up intellectual rights. You have your name written in a ledger. That's all "ownership" of crypto is: your name is written in a ledger, indicating your ownership of... nothing. 'Part ownership of x' could be used to describe anyone's stake in any Ponzi scheme ever.

"All investments only have value because other people think they do."

If you buy a 1% stake in a company, you own 1% of its assets. Physical, tangible things that have a capacity to do useful economic work. If you buy land, you can charge rent for its use, or occupy it yourself. If you buy a computer, you can use it to do work. If you buy an orange, you can eat the orange to metabolize its nutrients. In contrast, a bitcoin has zero capacity to do useful economic work; its "value" comes from the potential to find a bigger fool to sell it to for profit.

"Why hasn't all the money in the world been transferred to a few rich people already?"

All the money in the world is being transferred to a few rich people. This is what is currently happening. See Pixel Wealth.

"I think crypto has disproportionally benefited relatively poor people, who got in before most people with already established fortunes."

No.

https://www.wsj.com/articles/bitcoins-one-percent-controls-lions-share-of-the-cryptocurrencys-wealth-11639996204

approximately 0.01% of bitcoin holders control 27% of the 19 million bitcoin in circulation.

And to remove any lingering doubt:

1. No. https://www.newsweek.com/dogecoin-co-creator-says-cryptocurrency-inherently-right-wing-technology-1609862
2. No. https://www.fool.com/investing/2022/03/04/how-crypto-increases-economic-inequality/
3. No. https://gizmodo.com/bitcoin-crypto-bank-of-international-settlements-1849784466
4. No. https://twitter.com/davetroy/status/1478017698676228099
5. No. https://davidgerard.co.uk/blockchain/2021/03/11/nfts-crypto-grifters-try-to-scam-artists-again/
6. No. https://davidgerard.co.uk/blockchain/2021/02/12/libra-shrugged-chapter-6-banking-the-unbanked/
7. No. https://web3isgoinggreat.com/

1. How is Ethereum, in particular, a Ponzi scheme? How does new investors' money pay previous investors?

New investors' money pays previous investors whenever someone buys Ethereum from someone else. That's what buying crypto fundamentally is: a new investor is paying a previous investor. That's the fundamental building block of the crypto "economy". That's why they're all Ponzi schemes.

2. How would a bank run and collapse even be possible in principle?

It's happened many times already. The price of crypto collapses regularly, and then people who haven't learned the obvious lesson decide to "buy the dip".

3. "It is pretty similar to a publicly traded stock in this regard."

The difference is that a stock confers part ownership of a business, whereas crypto's value is entirely speculative; it's only valuable because other people think it is. The entire "value" of a cryptocurrency is derived from hype, like any Ponzi asset.

4. "It's not just open to the rich."

The more money you have, the more crypto you can buy. Thus, between collapse cycles, "investors" make profit in direct proportion to the amount of pre-existing wealth that they could afford to invest. In this way, speculative assets systematically transfer wealth from poor to rich.

Proof-of-Stake, in plain English, means that all holders of crypto are periodically gifted more crypto in proportion to the amount of crypto they already owned. It means the richest people are made richer. Meanwhile, the poorest people's crypto is reduced in value by the creation of all that new crypto which is being handed specifically to the rich.

If your solution to the energy problem is to aggressively worsen economic inequality instead, well, sadly, you've just replaced one problem with another. Or, even more accurately, you've replaced one problem by worsening another already-fatal problem, because the entire reason people buy into Bitcoin is that they hope to speculate on it as its value rises. This, of course, gives the greatest rewards to the richest people, because they were able to buy the most Bitcoin in the first place.

Proof-of-work crypto already drives horrific inequality, because it is simply a speculative asset. Proof-of-stake drives it far faster, because it is a speculative asset which multiplies itself, handing the new coins to the people who already had the most coins.

...Until it all collapses, that is, because it's a Ponzi scheme.

PoS does not alter the fundamental truth that *every* cryptocurrency is a Ponzi scheme (see the article I linked to in my previous comment). So honestly, all this talk I'm giving about the environmental and macroeconomic effects of crypto is laughable. It's a Ponzi scheme! Why would we need to have a discussion that goes into any more detail than that? Why would any of this other stuff even matter when we are talking about a Ponzi scheme?  Why does anything else even need to be said?

One obvious tangible solution: legalizing dense housing.

On top of all the ecological, economic, and health reasons why we already need to do that, low-density suburbia puts physical distance between everyone, preventing us from forming relationships. We could place special emphasis on housing projects that include communal hangout areas/recreation facilities, as well as old-fashioned community centers.