Not sure how that makes sense, if the stats change significantly then caches would be evicted during the gathering of statistics.

I believe popular connection poolers and clients attempt to do plan caching through prepared statements and keeping the connection open.

My understanding its not easy to do in PG since connections are process based instead of thread based and the query plans are not serializable between processes, so they cannot be shared between connections.

MSSQL has been doing statement plan caching for at least 20 years and it did stored procedure plan caching before that.