Battery won't charge, phone turns on when charger is plugged in

My iPhone 6 suddenly stopped charging, I’ve tried different power outlets and different chargers. My first guess would be that it’s either the battery or the charging port. However, if I turn my iPhone off, and plug in the charger, it starts up. This gives me a hint that the charging port probably works and it’s the battery that is broken.

Does anyone have any ideas? Prices aren’t that expensive for either the battery or the charging port, so perhaps it’s easiest to just change both?

Thanks in advance!