Is there any particular reason why you think using Aurora leads to trouble? If you have experienced issues with it in the past, please report them so that they can be examined and fixed.
Progressive Web Apps (PWA) are good alternatives to native apps. PWA are just websites, and you can install them in your phone directly from the browser. Try to visit eBay or your bank’s website, and see if you have a prompt that asks you to “install” the website:
If you are using Firefox for Android (I recommend Fennec instead, available on F-Droid), the option to install a PWA will be at the address bar, the house icon with the plus sign.
Note that this is only possible if the owner of the website specifically made it this way, but I think it’s worth a shot. Otherwise I fallback to recommend Aurora.
I wouldn’t recommend enabling root user in your phone, this is considered bad practice security wise. You can install something like NetGuard or TrackerControl, although I think only the pro versions will display real time traffic. I’m not really sure about this so I will let you look into it.