MacBook pro won’t stay on and has difficulties powering up

MacBook pro 13″ touchbar from 2016.

The MacBook sometimes will turn on, most of the time it won’t. When it does turn on, I can get as far as the login screen, where it will completely shut off after ~30 seconds. During that time I can see a battery icon in the top that says 1% and has a charge icon. SMC reset does not help. Getting into recovery is near to impossible with the spotty way it powers on at all.

The laptop has been stored in a cabinet for some time (at least 2 months), with unknown battery charge on storing.

Could this be a malfunctioning battery pack that no longer holds a charge and has not enough juice to power the MacBook? Shouldn’t the machine be able to run on AC power even if the battery pack is dead? Are there any solutions I can try?