So long as it's a reputable shop with newer equipment and someone who know what they're doing, you should be fine. It's hard to screw it up with a steel wheel. You can reduce the risk by pulling any weights off before you take it in- I've seen tires during removal get snagged by the weights and tear the bead. Seen as in been there- done that; I have my own new Coats tire machine and overlooked the weights. If the tire is dry rotted and going to break- it'll happen regardless of where you take it. I'd just make sure they know what they're doing and let 'em do it.
Did you see the size of that chicken?! -Dirty Steve