Why is the battery turned off for checking the voltage on the A320?
The voltage varies with what is connected to the bus.

Adding load will, for every electric source, reduce the voltage. And you wouldn't know how much is still OK without also knowing what is powered.
When the battery is charging (e.g. from ground power), you'll see the charging voltage even if the battery is empty.

With battery off you remove this variability, so they can give you exact number that signifies the battery is sufficiently charged.
(IIRC there are few things on the hot bus that are always connected, but their power consumption is known, so you check the battery under well defined conditions)