Doing so complicates other hairy scenarios because of the increased hydraulic/power demands, reduced clearance, increased drag on the plane, worse handling, and the fact that in some emergencies you want a gear-up landing and would expect pilots to simply make the opposite error (forgetting about the gear being automatically deployed) some fraction of the time.

Hmm. Are those situations more likely than forgetting to put the gear down? (Serious question -- I don't know the answer, and I'm curious.)

I think so, and they're definitely more severe. Gear-up landings occur in something like 1/150k flights, and they're rarely fatal. Stalls happen in roughly 1/100k, near-stalls more frequently, and if unrecovered then they are almost always fatal.

The one plane I know of with auto-retracting gear had a fatal stall because of the feature, so it's not exactly a theoretical argument, but there haven't been a lot of empirical studies.

You're talking about GA flights, not commercial ones, right? I do not believe that a gear-up landing occurs once a day on average in commercial airliners.

You do not want auto gear down. There are situations where having the gear down might makes much much worse and you might not be able to be in a situation where you can bring them back up.