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Did you mean "why doesn't the government allow every citizen to register a set of public keys for identification purposes"?

The citizens obviously wouldn't want the government generating their private keys. Because then it wouldn't be a private key any more.

Because securely storing the keys is quite difficult, and the whole system doesn't work if the keys are routinely compromised ("You could go in person to any government building and request a new private key to override the previous one if needed." doesn't cut it) - so the only reasonable way of issuing such private keys would be on a smartcard where it's reasonably difficult to extract/copy, and you could know if the actual card was lost.

Estonia has been doing this for a long time now with their ID cards. It can be extended to sim-based (mobile ID) or device/app based (smart ID) key pairs as well.

It's not just for the citizens but all residency cards and e-residency works this way, too.