Even if a full rework or overriding would be hard to implement, saving/loading statistics, or disabling the collector (after it already managed to do a decent run) couldn't be done? Or the nature/representation of the stats don't allow for this? (So they need constant updating, even if badly? But if stats only influence query plans then after good plans are found for the most common queries they could be persisted, right? Am I missing something?)
I think import/export of stats is pretty doable. Not a tiny amount of work, because of how many stats there may be (regular, extended), but I don't see any obvious major challenges ...
Similarly for disabling autoanalyze. We kinda already allow that for autovacuum (ALTER TABLE ... SET ...), so adding autoanalyze_enabled seems possible too. Or you could just set the threshold/scale_factor very high ...
