Secure communication implies three things: 1. the involved parties verify each other's identity, 2. communication is not compromised 3. the communication isn't spied on. The problem with self-signed certificates is that browsers can't establish (1), so they complain. Let's Encrypt really only solves the problem of browsers complaining, but it doesn't verify the domain owner's identity (eg. no credit card check).
I think document verification should be more stringent: you want to be sure whoever signed a document is who they claim to be and have the authority to do so, not just that they owned some domain and got a free certificate.
> Let's Encrypt really only solves the problem of browsers complaining
That’s not true — it proves that the certificate issuee (is that a word?) has control over the domain that the certificate is issued to, and thus prevents MITM attacks, which a self-signed cert does not do. It’s not simply a workaround to make browsers happy.
You are correct that it doesn’t prove legal ownership of the entity.
I think document verification should be more stringent: you want to be sure whoever signed a document is who they claim to be and have the authority to do so, not just that they owned some domain and got a free certificate.