I was not familiar with advice to pat meat dry, so I asked a chef what was going on. The answer was... complicated.
First, if you're working with meats that were frozen, you're going to want to remove any moisture which comes from the freezer (frozen humidity, which is essentially water).
For un-marinated meats, you want to be careful how you pat them. The important distinction, in all cases, is not to remove any natural fluids (blood) from the meat. This would remove both moisture and flavor, resulting in bland, dry cuts of meat. So, do not press the meat when patting; this will squeeze moisture out. You should lightly touch/brush the surface, so excess moisture wicks away. That's it.
For marinated meats, it can depend a bit on the marinade, and how much your supposed to keep for flavor. Otherwise, stick with the "remove excess moisture, but nothing else" idea.
So, long story short, if paper towels are sticking to the meat, you're removing too much moisture. Maybe try this the other way around: set a paper towel on your work surface, hold the meat in some tongs, and touch the meat to the towel as you're putting them on to fry.
As to how professional, large scale kitchens do it; they don't. They'll take a large sheet pan, place a drip rack in the pan, and place the steaks on the rack. This allows excess juices to flow off naturally, and everything is easily washable and reusable. If the meat is not cooked quickly after set to dry, you can baste drippings, marinade, or oil onto them just before cooking.