Opening most or all of them in MP and some UTG seems fine. If either blind 3bets a normal size and it gets back to us, is it correct for hero to never fold?

If calling closes the action and we are heads up versus one of the blinds, folding would be a mistake. If the raise is full pot and there is already a caller before us, folding becomes an option even if we close the action. If we are in late position and the pot will be 4-ways or more and calling closes the action, again folding becomes a mistake.

Of course if folding is ruled out, that does not rule out raising or jamming some portion of the time, in addition to calling.

Granted, in this spot we will be dominated more often so some actual skill will be required to take back the equity postflop that we give up preflop.

It depends on a few things, such as the sizes (of the open and the 3-bet, and remaining stacks), which of the blinds 3-bets (SB's range is not the same as the BB's) and the game format (including rake percentage), but I think you should fold some of the Axs.
If you always continued, your overall flatting range would either be too wide and weak (if you also called 55+, AQ+, JTs+) or it would lack board coverage (if you only called with something like 88+, Axs, AQ). If you flat an Axs combo, you usually want it to either be dominating a bunch of villain's 3-bet hands (AJs dominates a lot of Axs, but A7s does not), or to have decent equity against them (e.g. A5s does well vs KK).
The Axs hands I'd be most inclined to fold (or 4-bet bluff) would be A9s-A6s, since these have lower equity and playability vs a typical villain's 3-bet range. I'd rather flat with JTs or T9s than A8s, so that I'm likely to have 2 live cards, as well as be able to flop more draws.