If there were only 4 entities, your description would make perfect sense. It's a waterfall model, where the data only falls from Netflix to Level3 to Verizon to User.
But the Internet is a big hierarchy where things flow downhill. Every node can talk with every other node. Historically the networks would trade data freely if they were both doing an equal amount, but even 15 years ago I heard sober network engineers worrying that in the future there might be very unbalanced connections and what would be the fair way to deal with that.
NB: In the end I think Verizon is being purposefully difficult to their customers, but they are not completely unreasonable to think that having a lot of data sent into their networks [1] is just something they should STFU about.
[1] yes, even if you say "data requested by their customers" over and over again