Server will relay them unless "enforce-secure-profile" option is set to true in server options, in which case it will not let you join.

If I understand this correctly, this paragraph is about whether a 1.19 server has toggled an option in server.properties.

In that case, what is the message the client receives when trying to join it? Assuming it is a cryptic error message, could you add an explanatory one?

Enable it and try it out, it allows non-Microsoft accounts to join your server and players who disabled public key signing. You can see how this was implemented in a third party proxy that also enables this feature by default: https://github.com/PaperMC/Velocity/blob/1a1391a51916b111ceafe47bfbb144cf63cdc7ac/proxy/src/main/java/com/velocitypowered/proxy/connection/client/InitialLoginSessionHandler.java#L86

Enable it and try it out, it allows non-Microsoft accounts to join your server and players who disabled public key signing. You can see how this was implemented in a third party proxy that also enables this feature by default: https://github.com/PaperMC/Velocity/blob/1a1391a51916b111ceafe47bfbb144cf63cdc7ac/proxy/src/main/java/com/velocitypowered/proxy/connection/client/InitialLoginSessionHandler.java#L86

So, from that source it seems to use one of those vanilla strings:

Invalid signature for profile public key.
Try restarting your game.

Unable to parse profile public key.

Missing profile public key.
This server requires secure profiles.

That can get confusing with this mod, so I still propose changing this to something more explanatory.

In that case, what is the message the client receives when trying to join it?

By default - this:

That can get confusing with this mod, so I still propose changing this to something more explanatory.

Reasonable idea. I will.

Closing this since instead I will eventually implement warning-based proceed-or-leave screen, see #4.