Yes, you are right, it should work. I mean, if the battery is simply old, and its capacity is down, the computer, when plugged in, should work without any problems. But batteries also have a special control unit, which decides, when to charge, what is the current level, things like that. And that can also go wrong, and it may prevent the computer from getting electricity.
Can you get out the battery, and try to start the computer?
If it works, then something is wrong with the battery. If it doesn't work, then something is wrong with the computer.