A connection definitely has overhead in PG, but "5000 concurrent connections will already start to deadlock your Postgres server" is bogus. People completely routinely run with more connections.

Check the throughput graphs from this blog post from 2020 (for improvements I made to connection scalability):

https://techcommunity.microsoft.com/blog/adforpostgresql/imp...

That's for read-mostly work. If you do write very intensely, you're going to see more contention earlier. But that's way way worse with sqlite, due to its single writer model.

EDIT: Corrected year.