IMO the product is more of a factor. It's the product that does the work and is in contact with the paint. That said, the type of pad will add to it's cutting ability, but unless it's microfiber or wool, nothing that is going to make a huge difference.

In terms of your vehicle coming back from a body shop, it shouldn't require you re-compound it. I just had mine in (pure black) and it did come back with some minor swirls and what not from being "covered" and mis-washed but a quick hit with Essence was all the fully corrected panels needed. The painted panels I compounded as they of course came back in "body shop" condition (not detail oriented) and filled with holograms. Nothing a little Ultimate Compund and a purple wool pad didn't make short order of. I then polished and sealed the new paint with AAT505 which is essentially HDSPeed on steroids less any silicon thus safe for new paint.