I used to provide technical support for iPhones at Apple. This advice likely won't help at all (and can actually have the reverse effect.) Early iPhones/smartphones did have issues with multitasking and multiple apps open at the same time. That is when people suggested to "kill" unused apps. This changed maybe 3-4 years after the first iPhone launched, and there is no longer any need to "kill" or "close" apps unless the app is frozen. The software is designed to put unused apps in a saved state - and when you end an app, it has reload the app from scratch vs pulling it up in a saved state. I still hear people telling their friends to remove running apps in the background, but it is outdated information and only makes your phone work harder in the long-run.
From my playing around with the Disney app - if I keep the RoTR status page open, it does not auto-refresh. But if I tap the back button in the upper left hand corner and then click on "Find out more" - it immediately refreshes. This would be my recommended technique at open time. You don't need to reboot your phone, or kill the app, or exit out of the app. You just need to tap the back button and click on Find out more.
As for LTE vs WiFi...it's really impossible to tell which will give you better results. Towers and Wifi Nodes can only provide so much bandwidth in a given second. If there are multiple people in a group, I'd probably have one person on wifi and the other on LTE, just for a backup.
You do bring up a good point about iPhones but in general memory leaks are an issue for every device out there. It's less likely an issue on the iOS side given how good Apple has in terms of control of its hardware and software. Every Android version has at least had some memory issues and devices tend to be a hit or miss with certain updates. It won't hurt to do a shut down and reboot although it shouldn't really matter for most flagship devices, and if your device usually doesn't struggle with apps getting kicked out of RAM, it really shouldn't matter.
With that said I found your strategy for refreshing interesting. I actually used this strategy for my SO and I, except we are Android users (Pixel 3 and 4). One of us would go back between the main page and find out more to see if the join button would appear. I actually used the My status page where you could pull to refresh hoping the join button would appear when it was time to join.
Both of us had stellar LTE performance before 6am and at 6am our strategy yielded nothing. I'm also on AT&T while my SO is on Verizon so in some ways we had each other's backs. It wasn't until I heard boarding groups being shouted out I peered over my SO's device and asked why it showed nothing yet. We both simultaneously closed our apps and reopened them and voila saw the ability to join after clicking "Find out more."
This may be a difference between how the iOS and Android apps are coded, but perhaps on Android going back between the front page of the app and "Find out more" doesn't actually refresh your ability to join a BG or not. We ended up with BG65 and we were still done before 6:01am.
With only 1 data point, I don't want to jump to conclusions, but it could be perhaps our devices and networks were lagged, but I suspect it's more that our refreshing strategy maybe was the problem. While in theory you would think the app is refreshing everything as you go back and forth, it might only refresh the current BG status or your status, but not your ability to join a BG. It also could be the theory that some have suggested that there is some level of randomness in the BG assignments. People who finished AFTER us were still able to get BGs in the 30s.
All in all was super happy with the ride itself. BG65 was just an inconvenient spot because the ride got stuck around 61 for a LONG time in the morning and then broke down shortly after our BG got called again (Dec 28th)