Let me try anyway (maybe I'm a masochist)

First I'll say the government already has an ID system with a backdoor they mandate you use (your federal social security ID and state ID). The backdoor isn't very interesting because anyone with your ID in hand also has it.

So how about this:

1. State assigns citizens an ID at birth 2. State allows citizens to submit a public key along with their ID at any time 3. Citizens can go to their bank / private social network / whatever and say "this is my public key, you can use it to sign messages to me, and you can verify someone a) alive and b) a citizen of $state is reading it (from here you can bootstrap whatever protocol you want) 4. The state<>citizen network established in (2) is constantly under attack as stealing someones private key valuable so you also need a legal and technical framework to defend it

The protocol for submitting private keys and defending it from attack is a much longer post, I'm convinced there are ways to do it that drastically favor defense over offense, but that's not the point here.

Our question is can a government force it's way into the protocol you bootstrapped on top

How would they?

1. They could reset your public key to one they control the secret to, and then impersonate you digitally to break into your bank or social network. However I don't think they could do this secretly (the key update would necessarily be publically visible), so it's not really a back door. They can already do this with a search warrant. And if you're paranoid you can bootstrap your secondary cryptographic networks with multiple factors. So, this is on net more secure for you.

2. They could try to recover your secret key by force or warrant - but again not a back door.

I think the real concern isn't backdooring it's blacklisting, if this system becomes the L1 for every L2 crytographic interaction, they can practically remove your ability to freely transact. But that's a political problem you address with political means, I'm convinced from a technical perspective this is more secure and far cheaper for everyone.

Whatever clever crypto system you think of: if it needs to work for the general population, it needs to go hand-in-hand with UX.

Say your example: a user generates a pub/priv keypair locally and shares the public one with the government. How does the government know you’re rightfully sending the ID? How does the user know what they are sending? Can the app/website/tool/person at post office they are using to generate+store+send the public key be trusted by the user? How can the government give trust to the user that this tool/person can be trusted?

And there we have attestation again. Or walled app stores, or certification as we have for physical services.