I do not understand, average wattage/power would be integration of wattage (joules) divided by time (so back to watts), it’s still watts? Under what implementation would you end up with joules that isn’t just oddly broken in general?

I guess I could see a naive implementation (confusing integration/sampling and discrete summation) going the other way, erroneously ending up with a nonsensical W/s.

Also don’t understand what it has to do with eponyms, which are just substitutes for base units, either your DA works or not, no? Average wattage is kg⋅m²⋅s⁻³ not kg⋅m²⋅s⁻² (joules) or kg⋅m²⋅s⁻⁴