Tightening keyed drill chucks.

[Toolmaker51]

There is a hard to find chart that lists pilot diameter and tooth count of chuck keys. Some chucks [name brands] list correct key size. Some bevel engagement can be cured, lightly - minimally pressing the sleeve toward the chuck nose. Retract jaws and press with a short proper tube or bar so relieved.

My chuck technique is from an OLD machinist, used especially in large sizes.
Open the chuck sufficiently for bit. 'Rattle' the bit some while hand closing the jaws, helps insure small bit contacts all three, very helpful with hand-tight chucks. Then just as Tony emphasizes, lever position perpendicular to axis, I tighten at first hole. Not necessary to exert tremendous force, and never a cheater. You'll now be surprised as you proceed to hole two and three, exerting same force yet achieving more jaw grip. Provided jaws haven't spun too many shanks [usually taps], it will be far less able to do so.
Hand tight chucks work more on torque to hold securely, and they lose it reversing. So tapping [other than hand turned] is rarely successful.
I won't tap with a drill chuck other than initial start by hand, more or less like a tapping stand.