My laptop battery broke, so it is now flat.
Unlikely.
The built-in power supply should usually Correspond to the power consumption of the laptop.
Would be a blatant design flaw, right?
No. However, a laptop power supply unit usually provides sufficient power so that the laptop can be operated directly from the network (i.e. It works without a battery).
There also seem to be laptops that don't work at all without a battery. But I've never seen one like that.
No. Because a cable does not consume electricity in this sense (the power loss is too low).
A laptop without power will not work. Power is then either from the battery or via a power cord at the socket.
Incidentally, a defective battery should be removed. Because if you enter a defective battery, the laptop may also have no power.
Yes, the battery is removable, I took it out.
Yes, that can actually be the case. Especially if a non-original power supply (or a power supply for a "smaller" laptop) is used, it may be that the power supply does not have enough watts to supply the laptop with sufficient power.
With the original power supply, it should actually be sufficient. Of course, it may also be that there was an error in the design. It is best to check the values on the power supply with the rough power consumption.