https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kir...
The battery is either charging or discharging on net.
I sometimes see colloquialisms like "The laptop isn't really running off AC power, the AC is charging the battery and the laptop is discharging it to run". I also see people say that it's bad to keep a phone or laptop on its charger all day, and I usually do that to no ill effect, so I stopped trusting common wisdom on batteries.
When you get down to the cell level, each cell has 2 terminals, and it's either charging or discharging, right? Unless the charge circuit is doing some green-blue trick to charge half the cells and discharge the others, nothing that runs off a battery can be running AC "through" the battery. They taught us in A+ class that UPS' run power "through" the battery but I think it's objectively wrong. Especially if it's a lead-acid battery which I _know_ only has two terminals.
You can draw a boundary around the battery and charging stuff and say, if 90 watts is going in and the computer is pulling 80 watts, the battery must be charging at 10 watts. If the computer is pulling 120 watts, the battery must be discharging at 30 watts. But there is not 90 watts of power flowing "through" the battery in either case. Adding a load just causes it to charge slower, or discharge instead of charging.
It's probably true that the power supply only converts from mains to whatever the battery charger wants, and there's no path from mains to the load without going through the battery's voltage, but as other posters said, the reason you can't run some phones with a dead battery is more related to peak load. I have definitely owned mobile hardware that worked _pretty well_ without a battery, but worked better with a battery to smooth out load during startup or CPU load spikes.