Easy way to test that belief: just stop wiping and see if punctures increase.
I inspect my tires once in a blue moon while a bike is on the stand, and occasionally there is a tiny little thing to dig out of the tread, but punctures are surprisingly infrequent despite the amount of crushed glass and other debris that shows up on my roads. I suspect that wide, low-pressure tires shrug off little sharp things better than narrow high-pressure tires.