Does the sticky tape really hold well enough for helmet mounts?

YES! The 3M VHB tape has been used for literally over a decade by the biggest action camera manufacturer in the world for good reason. It holds extremely firmly with no wiggle and can be removed without damaging the helmet and without leaving residue. It's basically magic.

The key is just to make sure the helmet is clean, dry, and room temperature when first applying the mount, because obviously dirt and moisture can prevent the tape from sticking, and cold temperatures make the foam harder to compress to get good surface contact. Press it on there really firmly the first time and you will have zero issues with the tape holding.