Also check out ProxySQL [1][2], it's an extremely powerful and battle-tested proxy. Originally it was only for MySQL/MariaDB, where it is very widely used at scale, even despite MySQL already having excellent built-in scalable threaded connection management. But ProxySQL also added Postgres support too in 2024 and that has become a major focus.
And lets you rewrite queries on the fly. :)
+1 to ProxySQL, especially in RDS environments with huge monoliths attached that open a shitload of threads. RDS has fixed max_connections depending on the instance size so if you don't want to pay $$$$ for bigger but underused instances - and while you are trying to get the devs update all the hundreds old dependencies in the monolith to improve it, ProxySQL - can save your day. It did, for me. And yes, it's a self-managed system but it's pretty easy to operate and very stable.