It's still a pain to archive, and requires extra network roundtrips even in the best case. I not only need to be able to access my mail host, but wherever else the link is pointing to aswell.
When I'm checking my mail on my mobile phone, I'm sure as hell do NOT want to make extra network roundtrips just to be able to see my mail.
It's also far harder to index, sort and search in the proposed scenario: while I can quickly file an email under any label or folder I choose, and be sure that it will stay there, have the same contents forever, and I can search it easily, the proposed scenario does not allow for that. While I can still file and label the emails that have the links, I cannot easily search in the contents, unless I archived that separately, or I go online.
Again, a lot of indirection for no real gain. I can already use something like http://gist.github.com/ or the various pastebins, and a multitude of other solutions if I have a particular problem where they are appropriate.
But for everyday mail? Why would I want to do that? It just makes things more complex, for no purpose or gain at all in the vast majority of cases.