While 'Are today's durable goods too low quality / too disposable?' is a valid question, I think it's a separate one from 'How do we reduce, reuse, or recycle one-time use, disposable items?' You're not going to get everyone who comes to the parks to bring their own reusable travel mugs, so if disposable cups need to be provided in some form, what should that form be?
My understanding with the wax-lined paper cups is they aren't necessarily easily recyclable, nor compostable. Plastic drink cups could be a solution if they actually provide convenient recycling at the parks. There's a whole lot of trash cans at Disney and not enough recycling. Surely there could be some kind of innovative solution that accepts & diverts remaining ice/liquid while taking the plastic to recycle? Biodegradable paper straws like at Animal Kingdom solves the straw problem. Whatever the solution is, I'm okay with the approach being "how do we do disposable items smarter?" Fast food has been making strides over the past few years with biodegradable papers for bags, wrappers, bowls (ideally this should be paired with available compost bins, of course).