And yes, Dropbox does exist because of NAT, or more generally the fact that "consumers" are second-class citizens on the Internet. If everyone had a publicly routed address and an entry in DNS - which is how the Internet was originally envisioned - we'd have solutions that don't require a third party server.
Both of your examples just show how deep the Stockholm syndrome is. IPv4's limitations have shifted the way people think about networking so much that they rarely even realize they're doing it. But this isn't the way it was supposed to be. "Intended" is completely valid when we're talking about how something has strayed from its original design and the ramifications of it.
I'm not naive enough to think that we can ever get rid of the "consumer" and "producer" ideas that have crept into everyone's head. But it's important to remember that these ideas came about only because of limitations in the protocol, not because they're some kind of natural law of how global networks function.