The problem is arrival volume, not the form
An indoor playground does not receive guests evenly. You get a trickle mid-morning, then twenty families inside ten minutes when a school lets out or the weather turns. Every one of those families has at least one child who has never visited before, which means a fresh signature and a fresh set of names.
That is why paper collapses here in a way it does not at, say, a guided tour. A tour has a manifest and a departure time. A playground has a door. The waiver has to be finished before the family reaches the counter, or the counter becomes the bottleneck and the queue becomes the experience.
The fix is two-sided: a link that goes out with the booking confirmation or party invite so most families arrive signed, and one or two tablets by the door for the walk-ins who did not. Anything that requires a staff member to type a name is a step you will regret at 3pm on a rainy Saturday.