Why is electricity consumption billed in watts and not amperes?
They don’t bill in watts (power) either. They bill in watt-hours, that is, energy consumed. (kilowatt-hours typically.)

Let’s break this down a bit. Current alone doesn’t tell you power. You also need to know the voltage, as power is voltage * current. Then, you tally power over time to figure energy.

Now you could estimate power from current if you make an assumption about the voltage, and in the early days of electricity they did exactly that using ‘coulomb counters’, that is, they measured and tallied the current delivered over time and billed based on the tally.

This current-only method proved to be inaccurate because of line fluctuations, so for this and other reasons (notably, the adoption of AC power) they developed the motor-type meter, and later, the more familiar spinning-disc induction-type watthour meter. These meters also take voltage into account by design.

More about meter development here: https://www.smart-energy.com/features-analysis/the-history-of-the-electricity-meter/

And because it matters sometimes, a discussion of real vs apparent power and how utilities deal with it: https://www.electronicdesign.com/power-management/article/21806945/how-does-power-factor-correction-impact-your-utility-bill