So I've looked into these towel dispensers when I was learning about TENCEL and the company that makes it: Lenzing.

Companies in Europe can take the dirty towels from these dispensers, bring them back to their warehouses where they have massive drums for laundering, as restock hotels and businesses as needed. It's a pretty solid form of circularity.

Then, when those towels reach their End of Life (EOL), Lenzing has agreements with these companies to accept the cotton towels for use in their production of TENCEL. The final fiber ends up being maybe 60-70% TENCEL (twisted cellulose) and 30-40% recycled cotton. Then that fiber is sold to make clothing, sheets, maybe even more towels (one could dream).

Paper it definitely cheaper in terms of raw goods and processing, but you can't control the waste stream. Sure, you can have garbage bins nearby, but people can toss whatever they want in there. Having a machine run through the towels means the user doesn't have to think of care about disposal: only use. Really it's a form of extended producer responsibility (EPR) which is the holy grail of recycling imo. Plus cotton feels better compared to paper imo

Edit: another commenter spoke to the hygiene and convenience of it all. I agree that humans suck and so relying on a machine to work is a big ask given how little maintenance lots of places do. Shaking your hands dry is probably the most sustainable since you aren't using any materials