I'd test that but then my laptop wouldn't have anything to get power from.

And that is the entire point of running the test. You do not currently know for absolute certainty if the adaptor is the actual cause of the problem.
The purpose of attempting to run the laptop without the battery connected, and only the AC adaptor is to see if the adaptor is actually providing power.
Some faults with the motherboard prevent the adaptor from actually providing power to the battery's charging circuits, which obviously means the battery won't charge. The charger is otherwise able to function, but the fault lies with the motherboard, or maybe even the battery in that the battery cannot receive power to charge any more for its own internal fault.
Troubleshooting is all about testing isolating suspects until you figure out the most likely cause of the problem. No battery testing isolates the adaptor as the cause of the problem as a successful test absolves the adaptor as being the cause of the fault while an unsuccessful test leads to other possibilities.
A follow up test with a different adaptor will prove without a doubt that the adaptor is at fault on a successful test, while another unsuccessful test will point towards a motherboard fault.