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.

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.