I think just looking at equities here won't help solve the problem.

Some people will play very straight forward here. Some players will bet their entire range once checked to but shutdown once you c/c. Ofcourse in this spot it's a pretty easy c/c with KK.

Some players will bet their entire range, and barrel us often. We're then in a much more difficult spot, knowing we'll face multiple bets.

Some ppl will be suspicious about a check on Axx flop,and will check back. This could be ok, we may get to showdown cheap and maybe get one street of value on the river.

tldr: Decision to bet or check is impacted by both their calling range if we bet, and their betting range if we check (and how well we can play vs their tendencies)