I usually brew all-grain, but I do get the occasional 5-gallon extract kit. When I do extract kits, I put the ice in the wort, it works fine.
So it's true, commercial ice machines can harbor some gross stuff. But seriously, the makers know people put that ice in their kid's sodas at parties. So they don't want to risk getting their asses sued off by being too gunky. Here's an experiment, go grab a Q-tip from the bathroom, and swab under your water faucet. It's probably grosser than the commercial ice maker. And you top-off your beer with that water!?
Any nasties on the ice or in the tap water will be quickly out-competed by your yeast.