You don’t need your phone to be operational for the app to work. It’s just like any of the smartphone app platforms… you have a card on file associated with the account. The car is recognized when you plug in to charge, then automatically charges the account when it is used for the amount of charge you take. You can set it up on your phone, in the car, or on a computer, but only need the car to charge.
Tesla is rolling out a subscription for $13.99 a month for up to 5 charging sessions per day, though. I’ve been traveling a lot and have been charging 3-4 times a day for 50-60$ total. It’ll be nice to save even more money with the sub, but that wouldn’t be worth it if you don’t travel a lot.
I’ve had mine for 4 years and, up until this year when I started traveling, I never used a public charger. Charging at home was always cheap and easy and I never drove more than 250 miles in a day. I’ve been pleasantly surprised at how easy route planning automatically plans the most efficient route with stops to charge along the way. I had a bit of range anxiety when I started out, but that was mostly from my ignorance of how it would all work out. Lately I’ve been driving through the mountains in winter and haven’t had any problems.