Massively inefficient compared to something like the certificate authority system, and you could have a "certificate transparency log" like system if needed.

You don't need a blockchain to do signatures or other cryptographic operations, and in fact the blockchain is the worst way to do it if you do not have adversaries trying to make double-spends.

It's much harder to write correct code if you have to give equal power to all peers, while also trusting no-one.

It's much easier to do what most systems do, which is trust every peer more or less (aka the SMTP style usual federation), or trust a small number of well-known supernodes (aka the CA system, the DNS system, the BGP system, etc).

All of those systems for distributing information of various kinds are vastly easier to reason about, operate, etc, than blockchains.