Switching to a threaded model would be a lot of work, but there is a simpler solution that retains most of the benefits: using a process-per-connection model for active connections only, and allowing a single process to have multiple idle connections.

I follow the mailing list because I'm interested in this exact issue. Konstantin Knizhnik sent a patch implementing a built-in connection pooler in early 2019 that uses a similar approach to what I just described. The work on that has continued to this day, and I'm hopeful that it will eventually be merged.