I don't think that FDIC insurance is appropriate in this case. FDIC is intended to protect the deposits of customers when a bank fails. Even though there are already many kinds of investments available through banks, the insurance only protects deposit accounts, and I don't think that will ever change.

On the other hand, SIPC is a similar kind of insurance for brokerages. Like FDIC, the insurance protects you if your investments (including Bitcoin ETFs) are held by a brokerage and the brokerage goes out of business or the investments are lost or stolen. I think this is what you are looking for, and it already exists.

What SIPC Protects

Anyway, I don't think that this kind of insurance would have any significant effect on the rate of adoption or the price, since the asset itself is not being backed.

BTC is not backed by any government and that cannot happen, BTC is backed by its features, like decentralization, scarcity, etc. BTC is not centralized, so it cannot be backed by a centralized government.
Self custody is not a burden, that is how BTC was designed to be used, with the owner being their own bank. If anyone sees self custody as a burden, i would not be wrong to say that they do not really understand what BTC is.

If government distributed all their bitcoins to us, and took all the money rich have, and gave it to us, and if medicine never cost a dime, and if free food dropped down from the skies that would be a great world.

However, we live in a real world and sometimes we need to know how much we can safely assume could be done. Preferring bitcoin to be insured like that is a big ask, considering the volatility of it, we are talking about a huge deal here that would ruin a governments accounts and treasury so there is no way that any sane nation would do something like this, specially when they know it wouldn't bring them that much vote to help offset the people who will hate losing money because of this since it would cost a lot.

Even if this were to be implemented, I would personally be very skeptical of whether or not a government agency would be able to provide a seamless claim process if a bank were to ever close its doors and those who held Bitcoin within were at a loss. If a bank were to mismanage or in some way "lose" the Bitcoin, to purchase the amount of Bitcoin that bank held would cost the government a hell of a lot of money (probably more than the market value) and it would need to buy it on market. It would probably cause an economic spiral (depending on the amount of Bitcoin stored), as the money would probably need to be borrowed (printed) too. This is also not to mention if the bank was even holding the Bitcoin reserves to begin with, and if what customers held were just Bitcoin on paper.

People should learn self-custody, not create frameworks and rules that only further reliance on banks. After all, one of the main points in Bitcoin was to reduce reliance on banks (arguably, to eliminate them altogether).

That is really interesting way to think about it and you have made good point about why having government FDIC insurance for Bitcoin in banks could be big deal maybe even bigger than Bitcoin ETF. You are right that Bitcoin ETF might not be safe in long run because future leaders could change rules. Just knowing government backs it even if they do not spend any money on Bitcoin could make huge difference in how people feel about it. You are right that kind of trust is hard to beat for many. Giving up control of your own Bitcoin is big thing and it is key idea for many Bitcoin fans. But you have explained well why some people might prefer ease and safety of FDIC insurance over having full control. Comparing it to ETFs and companies like MicroStrategy shows how this could be simpler and maybe safer way for big and small investors to get into Bitcoin.
Idea of getting loans from regular banks using your FDIC insured Bitcoin as security is also really interesting. It could make it easier to get money and connect Bitcoin more with traditional financial world so people might not have to rely as much on riskier online lending systems. It is practical way of looking at it knowing that not everyone wants to deal with responsibility of holding their own Bitcoin. While keeping your own Bitcoin is still core belief for many of us so they will always deny it.

A future president or Congress can dump FDIC insurance as well. Anyway, your idea won't happen in real life, so there's no point discussing this.
I don't get the people, who are like "let's put our BTC in banks and let's make the government guarantee those BTC". Wasn't Bitcoin supposed to be anti-bank and anti-authority? You are basically the same as the Strategic BTC Reserve supporters, who want the US government to buy BTC and pump the Bitcoin price. FDIC insurance can bring trust and stability in the crypto markets, but what happens when banks start going bankrupt? The US government can't pay trillions of dollars in FDIC insurances. The Federal Reserve will have to print trillions of USD and solve the problem.

Those who are not ready to be their own bank should have nothing to do with BTC, and least of all trust that banks will give them the opportunity to secure their investment in Bitcoin. Technically speaking, any person with an average IQ can acquire sufficient knowledge about secure storage and how to make and receive transactions in BTC in a very short period of time.

Instead of teaching people to accept BTC with the features that make it unique, some are looking forward to it being equated to a fiat deposit in their bank account. I always say "be careful what you wish for, because it might happen".