The posts could be right about the processor overload, rather than bandwidth itself. Something to try in the interim is to block OS updates during service times. We have done that and it greatly relieved bandwidth and firewall load. We have an NSA 3600 with 50 x 50 fiber internet, and up to 500 devices combining wired and wireless, and still have good performance.
You could have 100 or more phones start running OS updates when arriving on the network.
Another thing, we installed Sonicwall Analyzer to assess network usage. If your pastors think everyone is sitting there using You Version, I have news for them. Snapchat is our 5th highest user of bandwidth on a typical Sunday morning. We also block proxy servers with firewall app rules to prevent users bypassing our filter. The wireless controller also turns off the wifi during the night, so someone does not get on public wifi during the night and try causing us trouble.