" their natural response would be to climb a fast as possible"
No, unless you are at an unsafe altitude, below the charted MSA.
Pilots should be trained when they regain control, to put the plane level, at a safe altitude/speed, and possibly on a safe route - assess what went wrong, plan what to do next, and inform traffic control - which will help to clear the path from other airplanes whenever needed. For example, if you plan to return to the departing airport, it's usually useless to gain altitude, unless you have another good reason to do so.
Of course pilots can do mistakes, panic, and follow the "natural" response instead of what training should have taught them. Still, it's possible that the different handling characteristics of the Max for which pilots weren't required to train for, may deceive pilots trying to recover the plane even after the MCAS has been turned off, even if they don't try "to climb as fast as possible".