Exactly, if the demand isn't/can't be flexible (eg healthcare, housing) then no matter the amount of competition on the supply side will balance things out in an unregulated market in the long term.

Everyone with the freedom of choice is on the supply side, and everyone on that side benefits from higher prices.
And there is no other metric of success other than profit.
(Nobody has the incentive to undercut the competition unless for later monopoly reasons. But the same manoeuvre can be achieved cheaper via mergers without the loss of revenue due to lower margins.)

That's why healthcare was a problem (with a constantly giant inflation) in the USA even before all the consolidations.