I guess Paperpile makes thing so simple that we get easily spoiled 
Two clicks is pretty good.
Very early on we used to have Paperpile buttons in Google Reader because it makes sense to integrate with something so popular. Now the user base is fragmented over many other programs and having an integration does not make sense.
At the moment the import goes through the extension and does not offer a public API. We might add a way to post a link via the public URL as you suggested. Also we have concrete plans to allow mailing papers to Paperpile which should make it possible to integrate with pretty much every software that has a share function.