That would make me incredibly irritated too! Kudos on your proposed plan. It makes sense.
My school actually does something similar to your proposed plan. I used to despise the GPA system, but now that I see what you mean, I definitely see the merits of it. For privacy’s sake, I’m not going to reveal the name of my school, but it is a large public high school. For us, we calculate unweighted GPA using the normal 4.0 scale (with an A+ meriting a 4.33). AFTER the UW GPA is calculated, they add ( (number of honors credits)0.096 + (number of AP/dual-credit credits)0.143 ). I don’t know where the specific numbers of 0.096 and 0.143 come from, but using your example above using Jane and John, their GPAs would turn out like this:
Jane (with band): 4.286
John (with non-credit study hall): 4.286
So, in answer to your question, yes, I have seen a school (my school) this calculates GPA and rank this way. And yes, I agree that it would be more ideal if your school did this too.