To my knowledge the current still goes through the battery in laptops. But I could be wrong. And I wouldn't know if that by itself wears more on the battery (other than the heat, if only for being just next to the hot laptop).

Regardless batteries found in laptops don't like being 100% charged for prolonged periods, just as they suffer if you leave them at 0% for prolonged periods. Also, you need to condition them.

And added to that an old battery. Typically I'd say that a laptop is considered old at the earliest about 2 years after its purchase, that's when the battery wear typically (although greatly dependent on usage) starts to really show and the deterioration starts to excel.

You really shouldn't trust even a new consumer laptop battery, but an old borderline to depleted battery that runs 24/7? In a home?