Hair and lint can stick to the ice, but try it for yourself, ice isn’t a sponge. The only way for anything on the surface to work it’s way in is to melt it’s way in, and then freeze the outer shell again. Akin to dropping the ice, kicking it under the fridge, fishing it back out, tossing it back into the ice tray. In which case, you deserve all the hair in your cocktail.

I’m actually not joking, if you rinse an ice cube. Superficial ice immediately melts and is rinsed away. You could have dipped it in engine oil and it would be immediately pure ice after rinsing.

Rinse the ice and then keep using it. It’s literally pristine again.