Yes -- he would only have to worry about setting up and maintaining a WebDAV server. Plus, he would have to do without integration with tons of his favorites apps in desktop and mobile (like Dropbox has).
You can save to your WebDAV using the standard file picker. Surely an app that exports to Dropbox also gives you a "Save As" dialog.
Except for every iOS app that integrates with Dropbox.
Very true, iOS doesn't give users control of their data. To me that's more of a reason to not use iOS rather than a reason not to use WebDAV.
"You shouldn't use Dropbox because WebDAV is better."
"Sure, but that doesn't work with my iPhone."
"You shouldn't use an iPhone either."
"WebDAV should be enough for everybody"
But then it won't work offline
You can also avoid syncing too much data with a little organization, by specifying which folders should be synced to which devices.