Well, if the battery is too hot, just gear down.
Seriously, no matter what, drilling fairly large holes into wood is gonna take some power. Even though the adverts say "heavy duty" or whatever, they apparently are not truly made for that application. Gotta figure the power requirements too. A corded drill that pulls say, 5A @ 120V AC is 600w. To get that same power out of 18v, you need 33.3A. A 2.5Ah pack will only last 4.5 minutes. Reduce that runtime further by the higher loads. Not to mention you need a continuous C rating of 13.32, and that's assuming a 5A draw - it could be higher.
Just like BL, HV is king for high power, which is why 120V excels.