This sounds really interesting. Query planner acquires lightweight lock on every table partition and its indexes. There can be only 16 fast locks, exceed that number and locking becomes significantly more expensive slowing down queries significantly. Workaround I have used so far was to use plan_cache_mode=force_generic which prevents re-planning queries on parameters change.

This approach might be a better option, but sadly app needs to be modified to make use of it.