After reading your original posting in more detail, I see that you
understand that to some degree.
The alternative method is: multiply each average by the number of items
averaged (the original divisor), add all of those products, and divide by
the total number of items (the sum of the original divisors).
Without translating that into a proper format -- it can be done, but it
seems unnecessary -- let me just comment this way....
Yes, that is close to the correct alternative conceptually.
But I do not see the point of it since, without helper cells, you must do
COUNT(B2:C3) as you noted. And that is no less trouble than AVERAGE(B2:C3).
In fact, the whole thing is a lot more trouble.