Why is it a bizarre unit? I'm billed for my electricity use in kW/h.
No, you're billed in kWh.
In SI, energy is measured in joules, and power is measured in watts. One watt is one joule per second. It's indirect to say "watt-hour", because in effect you're saying joules per second times hours when you could have just said joules.
That's true. I just think of kilowatt-hours as a single unit, like Joule. That way it's no more bizarre than saying Newtons instead of kg m/s^2