Joseph Pilates built the method around the feet.
Joseph Pilates developed his method, originally called Contrology, around breath, concentration, control, precision, alignment, flow, and full-body awareness. He also created the apparatus to serve that method: the Reformer, Cadillac, Wunda Chair, Ladder Barrel, Spine Corrector, Foot Corrector, Toe Corrector, and Ped-o-Pul.
The feet were always part of the system. In Pilates, the foot connects the body to the floor, the mat, the Reformer, the Cadillac, the Wunda Chair, the Megaformer, every surface beneath you. That connection is the work.
Pilates depends on feedback. The foot has to sense, press, articulate, stabilize, and respond. A grip sock that slips, shifts, or bunches moves your focus off form and onto your footing. It is like holding a plank on a sliding towel. You can still do the move. Your body is compensating instead of engaging.
Clean feet, clean floors, wiped-down equipment. That's what creates a clean environment. The tool under you should support the practice, not work against it.
If you prefer a sock layer, have cold feet, or train in a studio that requires socks, Performance Skins can be worn over a very thin pair (toe-free or fitted) so you still feel the grip.