We can set different rules on these networks however. We can choose to be choosy at the gate.
I spend about 30 seconds thinking about this, so this is clearly the perfect solution with zero drawbacks or tradeoffs.
A CDN. What you are describing is a CDN. We have CDNs today and the problem still exists because most of today's websites refuse to operate within the constraints. There is no need for new infrastructure to deploy this solution, we just need website operators to "give up" and operate in a more static way.