The way I view it, the company is not obligated to provide ice in the same way they are not obligated to provide breakfast burritos or hotdogs/hamburgers. If they choose to give you any of those things, feel free to partake off the clock.
If they are asking not to use the ice machine while on the clock, and this bothers you enough to make a stink about it, then remember they could take the ice away whenever they want.
It does seem kind of petty for them to care about only what should only be a few seconds. But I've known guys who seem to always have to poop every day on the clock right after start time. When you're on the clock you should be in service to the employer. I could go in to work and once start time hits, take off my boots and say my toenails need to be clipped due to safety reasons. But I should have done that before I got in.