We have done this with uniforms already. Uniforms have been huge logistical nightmare for us for years with inventory, fitting, delivery, storage, etc. being massive headaches and thousands of man-hours of work over the years. Now our awesome vendor is doing the majority of that for us. There have been some understandable hiccups the first year as we switch that process over, but we are able to spend much more energy on the cheer stuff, which has been great. (Customers just cannot begin to understand how big a challenge getting uniforms, practice wear, etc. designed, prototyped, redesigned, approved, fitted, ordered, produced, quality checked, shipped, inventoried, re-shipped, counted, labeled, distributed, re-fitted, returned, etc for fluctuating rosters and growing children can be.)
I doubt the EPs want the craziness of accounts receivable and collections for all of the athletes. They just want one big check/payment wrapped up in a tidy bow and let the gyms deal with chasing parents down for all of the entry fees. We already have to chase down tuition, so entry fees isn't really that much additional work. (You would be amazed at how many people don't pay and then expect to get their service anyway.). In fairness, gyms often delay payments to EPs as well, but this is likely because the entry fees haven't all been collected and gyms can't essentially "loan" the athletes the money for entry fees hoping they pay later.
I don't see how there would be any tax savings, to be honest. I'm not sure how you mean that. Pass-through expenses like entry fees don't incur additional tax. If you collect $1000 in entry fees and pay out $1000 in entry fees, there wouldn't be any net income to tax.
There would be savings for the gym on credit card fees I suppose (2-3% typically) , but that would just likely be added to the EP side.