Looks like it's started to work already, at least with Google, at least for me. I notice that Google now links to the version 14 docs for certain likely-common search terms, like "create table postgresql".
I am also now seeing the version 14 documentation for many similar searches on Google, and specifically the /current/ URL, which makes me think it is a direct result of this change. For example: "pg_stat_statements", "auto_explain", "postgres create index", and "postgres explain analyze" all now link to the /current/ URL too.
There are some that still link to old versions (again, for me, on Google), including "postgres create view" (to version 9.2), and "postgres gin index" (to version 9.1). I'm pleasantly surprised how many have already flipped though.