Since it fell, I’d say you have some hardware damage, more specifically the USB C port, even more so since you say it isn’t the first time. It will more than likely have to be changed, something probably got bent, or maybe the solder connection broke, that’s why you have to manipulate it to get it to connect. You’re probably still missing one pin/connection to actually charge the battery, almost as if you’re only getting the “DATA” connections. USB C connectors arn’t the strongest thats for sure, and to snap the cable…it definitely took a hit.
It could also be an IC, controlling the charging, maybe it got shorted when the computer fell and the cable snapped.
I assume you have you tried charging from both USB C ports?