Won't turn on unless plugged in, but keeps running fine if unplugged

My laptop has this strange issue where it will refuse to turn on - zero response from power button - unless the charger is connected, but once it's booted up I can disconnect the power cable and it will keep running on the battery just fine.

Also, I can restart it and it can wake from sleep just fine. It just won't power on again if I shut down, unless I 'jump start' it with the power cable.

The battery is from 2016 but the capacity is 90% and it lasts for about 10 hours with light use so it seems to be in good condition.

Any help in understanding this would be greatly appreciated.