chore(proxy): demote more logs during successful connection attempts (#9828)

Follow up to #9803 

See https://github.com/neondatabase/cloud/issues/14378

In collaboration with @cloneable and @awarus, we sifted through logs and
simply demoted some logs to debug. This is not at all finished and there
are more logs to review, but we ran out of time in the session we
organised. In any slightly more nuanced cases, we didn't touch the log,
instead leaving a TODO comment.

I've also slightly refactored the sql-over-http body read/length reject
code. I can split that into a separate PR. It just felt natural after I
switched to `read_body_with_limit` as we discussed during the meet.

proxy/src/control_plane/client/neon.rs

@@ -380,6 +380,7 @@ impl super::ControlPlaneApi for NeonControlPlaneClient {
         // after getting back a permit - it's possible the cache was filled
         // double check
         if permit.should_check_cache() {
+            // TODO: if there is something in the cache, mark the permit as success.
             check_cache!();
         }