I did that on my Honda kicker leg. I tried and easy out, then broke it off. I was able to take the midsection off the motor and take it to a machine shop with an EDM (electrical discharge machine). It's runs a current and slowly pulverizes the broken bolt out without damaging the aluminum threads. It wasn't very expensive, definitely less than I wasted on easy outs, harder drill bits, left hand bits, and for you, taps. I try to remember to loosen. lube and retighten the bolts every year now even if I don't need a new impeller.
I don't know if the machine could get at your housing if you trailer the boat in.
I imagine my thermostat is in there good, since I have never changed it.