Yes, DNS is good enough for organizations (though I should note that governments don't rely on DNS registrars, they run their own). The crucial difference is that organizations can hire people to care about those things, or hire lawyers to ensure they can be recovered if accidentally lost. And in fact many of these organizations change domains in the longer run for various reasons. So even there, DNS is often only used as a solution to identity at one point in time, it's not meant as a permanent solution for the entire life of an organization.
For individuals, the cost of losing your domain is far too high if it means losing your identity on multiple services at the same time. And, if nothing else, people eventually die, so domains will be lost by their original owner and then re-used, breaking the notion of identity again in the longer run.
There can be two kinds of identities: your actual legal one and ones you should be okay with losing.
People keep devising more and more involved ways to maintain identities other than your legal one, but if you think about it you can still lose any of those ways (your domain name, your private key, etc.) and in the end no one should use them for anything serious.
Identity also needs to be hard to copy (which is the main reason e.g. you would not want to use your a hash of your DNA as an identity -- it would be hard to lose but easy to copy).
How hard is it to run a registrar? I’d assume there are decent open source components to manage it. It’s basically a key value store. Why couldn’t governments provide a stable domain name for each individual?
For individuals, the cost of losing your domain is far too high if it means losing your identity on multiple services at the same time. And, if nothing else, people eventually die, so domains will be lost by their original owner and then re-used, breaking the notion of identity again in the longer run.