There has to be a logical explanation for what's happening.
Whatever happens with the Toyota app is probably unrelated. That's using the Internet and the cellular data module in the car to communicate, not the same system as the key fob.
What else is going on when you're testing? I know unexpected things will happen if both key fobs are within several feet of the car, or if at least one key fob is inside the car. Also, I know the remotes will behave differently if the engine is running. Haven't tried it yet with the 2020 HL, but with my last HL and my Tacoma, it always drives me nuts when the engine is running, I'm outside of the car, and the passenger doors are locked. The unlock button on the key fob won't unlock the doors in that situation.
When that battery-saving function is activated on a key fob, all that should happen is that you cannot unlock the doors by getting that fob near the car and grabbing the handle. Since you're able to unlock the doors by grabbing the handle, I think it's safe to say that at least one fob is not in battery-saving mode.The buttons on the key fob should continue to work when pressed.
Another thought... after you unlock the door one way or another, open at least one door. If you command the doors to unlock, they will lock themselves again after 60 seconds if no door is opened. (That time interval can be adjusted in the menus on the 8" or 12" touch screen.)