Foamicate uses a browser extension. Is there any reason the client side portion can’t be implemented in JavaScript with keys stored in localStorage?
If you consider it a bad and a security risk, then I would answer that the RSA private key would still be protected by a master password on the client side. You essentially end up with a solution like LastPass or 1Password with a different password for each service, with the added benefit that the server only has the public key so users aren’t affected if attackers obtain a copy of the server database.
If a good thing, well, it’s not clear to me that Foamicate syncs across browsers either.
There's a feature request from a year ago: