The best reason I can think of is portability. Visiting your parents and using their computer? No need to install a native app, just open the web page (hopefully in the future at least, this says you still need to install a plug-in). Want to use it on Linux? No need to wait for Microsoft to update the Linux app (if they ever bother to update it), just use a standard-compatible browser to open the web page. Microsoft wants to add features? No need to make sure it works on a bunch of different OSes and versions, just make sure the web page is still compliant with the standards (you still need to make sure the browsers handle the standards correctly, but this should be an easier target).
Not that this is perfect by any means. But nowadays computers perform well enough that most users won't even notice the problems you mention, and the other advantages more than make up for the problems.