There is already an extension that allows you to upload extensions, and there's another extension that allows you to upload themes. The only draw back of course is that you have to manually install these extensions first, then all subsequent extensions can be browsed to with the extension and uploaded automatically through the customize tab.
I can't remember their names off hand now since I don't use them anymore as I am accustomed to manually installing extensions, but it might just be called "Upload Extensions."
It would be nice to have Wordpress-like features for phpBB though. But even with Wordpress, you have to drag and drop a plug-in to the FTP directory folder and then go into your WordPress site and activate it. But the core update process in WordPress is automatic. Or you can opt to turn that feature off which might be a wise idea so that you can see if a new update creates unforeseen issues, and if so you don't have to auto update with one click until the next patched version comes out.
With my WordPress site I allow automatic updates because for the most part I haven't had any troubles or plug-in incompatibility issues, etc thus far. When phpBB 3.2.2 came out I think it was, there were at least four files that needed to be hand edited. So that's why I would be a little reluctant on having phpBB auto update without making sure there are no unforeseen issues. I like to wait at least a week or two when phpBB updates comes out and see how everyone reports back.
But I have in place now a mirror copy of my forum who's index is protected with a basic HTTP password to keep search engines away from it, etc. In here I test things and new updates. If all goes well then I know I can update my main forum. Which brings up something. If phpBB does come out with an automatic core update facility, I hope it allows you to click an update button rather then doing things behind my back without my permission. Especially if my theme hasn't been updated for the latest phpBB version. I always wait until my current theme has been updated before I move to update my forum.