Question - Could an embedded scripting interpreter at the same time be a language server?

This sounds similar to SLIME (and its fork SLY) for Common Lisp. These tools use their own protocols rather than LSP in part because they pre-date LSP but also because LSP is not a good match for everything they do. In addition to the usual LSP functionality like symbol lookup, xref and completion they also provide a very good Lisp REPL. It is easy to have the server side (called swank) running as part of an existing service which sounds a lot like what the post describes. For example I've used SLIME to connect to my live StumpWM sessions.