The reason that battery capacity is measured in amp hours is that you can't know the exact voltage of the battery as it drops with lower levels of charge.

Newer large-capacity battery systems seem more likely to specify battery capacity in (k)Wh. I think this is because they come with sophisticated battery management systems that keep the battery within a narrow charge (20%-80%) range, so therefore they can be assured of the bounds of the voltage.

Also, Li-ion and LiFePO4 batteries have a much different discharge curve than earlier chemistries [1], with much more constant voltage up to 80% discharge.

1. https://www.powertechsystems.eu/wp-content/uploads/sites/6/2...

If voltage drops with lower levels of charge, then so does the current (assuming a load with a constant resistance). Isn't that just another reason to use Watt hours instead?