That could be done; WebDAV introduced several new methods to HTTP. The drawback is that your clients lose the ability to use standard HTTP libraries to talk to

That could be done; WebDAV introduced several new methods to HTTP. The drawback is that your clients lose the ability to use standard HTTP libraries to talk to your service because you're using a non-standard set of methods. The 'Uniform' in URI and URL doesn't just apply to the resource identifiers; in a RESTful system the methods and representations both need to be uniform as well. That allows general-purpose clients to be able to interact with the service, and that was what made the web take off when earlier FTP and Gopher systems did not.