@KnightRiderKit Sorry for the jargon.
Walking up to the car, the phone connects directly to the car via Bluetooth radio to unlock it. This is when having the app running can help because the phone saving battery power by sleeping or by keeping the app asleep will make it take longer to wake up and connect to the car.
The "notification shade" is the bar at the top of the phone's screen where you can pull down to access notifications like new emails. Is the Tesla "T" in that bar? When you pull down to see notifications, is there one showing the Tesla app, listing your car? This indicates the app is running in the background and able to connect to the car.
Inside the Tesla app, it has features like viewing the car's state of charge %, operating Climate, and Controls like opening the charge port. These features work whether you're near the car or far away ("distant," "remote"), as long as both your phone and your car can connect to the internet via WiFi or cellular data.
A more expensive phone is not likely to get a more reliable internet connection unless that gets you 5G. Switching mobile service providers is another way that might help.
You can improve internet connections in the garage by moving your router closer or installing a
WiFi repeater or extender. Yes, moving the router could impact the computers in the house.
To evaluate those possibilities, I'd take the phone around the house and look at the WiFi signal indicator or at an app like WiFi Analyzer. You might want to get a friend to help you work through these tests and possible changes.