You would think that the NFL could design helmets that don't have grids to reduce face mask penalties...  Can't a multibillion dollar industry replace the grid w/ a clear alloy or composite that is harder to get your fingers into?

Yes they can and yes they should.

It looks very tempting to grab, but I'm guessing illegal?  You could snap a neck and paralyze someone, twisting their head around like that.

Very illegal to discourage that, but as Matt says, why don't they change the design?