Why do most HTs have 2m and 70cm as the two bands? Is it just out of history or tradition, licensing restrictions, or is there something in the transceiver or antenna design that would make these the two obvious frequency bands of choice?
History and tradition are good choices of words, yes. Historically, they have been the most popular bands for mobile work; the antennas are small enough for a vehicle and many repeaters have been constructed for that reason.
6 meters used to be a popular mobile band back in the 50s and 60s. (For that matter, so were 80 and even 160.) All those bands required a larger antenna, and back in those days many hams made their own. And since 2m and 440 were so popular, 220 never caught on except in some terrestrial and EME circles.
One reason 1.25m / 220MHz is not too popular is that not many other countries than the US have allocations in it, just like we (the US) do not have the 4m band.
Other reasons including what was previously said, could be size/practicality of antennas and useful RF distance. 6m has better propagation and bigger antennas, 23cm smaller antenna/more manageable but quite a bit line of sight propagation. Everything is a trade off in RF.
Virtually any HF antenna on a mobile is a compromised antenna by using coils to make it electrically longer than it physically is, which reduces its effectiveness, but there are those that do it because they can.