The MacBook of a friend broke and we are trying to fix it

The problem was that it always showed low battery. He took it to the Apple Store but it cost too much.

They said it was the motherboard he asked me for help to buy a motherboard and replace it. We bought one but it still has the same problem I tried booting pressing the shortcut to turn on without battery and it boots without it. But the fan was working very fast processor heat a lot and also no integrated keyboard or trackpad work.

Any help? Could it be that the Apple Store was wrong and it was not the motherboard?

Tried to turn on in diagnostic mode but when it run it just appears a black screen and they mouse but you can’t move the mouse