The primary aging effect in a Lithium-ion battery is increased internal resistance (caused by oxidation of the plates). This doesn't affect the Ah capacity, but it does reduce voltage and waste power at high current. Since voltage also drops as the battery discharges, the increased resistance causes it to reach cutoff voltage earlier and so reduces its effective capacity.
An old lithium-ion battery which is not powerful enough to run the device it was designed for may still be useful in a lower current application. General Motors and Nissan are reusing old electric car batteries as stationary storage for homes and businesses. At the lower current drain required these 'worn out' batteries can still deliver more than 80% capacity.
Using a power pack on an appliance with an old Li-ion battery will not use any more power than normal. The power pack will simply take over from the internal battery to supply the power the device needs. This is not the case with with old Nicad and NimH batteries, which tend to become leaky as they age and require constant topping up, which does waste power.