You made some very interesting points that I hadn't previously considered. Flash did create a richer environment with which to develop, and was ahead of its time in giving developers more powerful tools with which to craft more interesting and useful web applications. Its success drove innovation and ultimately spawned the HTML5 standard.
Because of the way flash is distributed, as a browser plugin, it allows for accessing OS resources that would otherwise be inaccessible. This is a big use for flash still today, as an ugly but needed bridge over which one can reach the camera and microphone of the host.
Flash began the trend of extending the browser's functionality further and further into territory that used to be reserved for native applications. A trend that continues today with the slow and painful adoption of WebGL, websockets, the canvas api, the A/V streaming, etc.
The big advantage that flash still enjoys today is that it presents ONE working interface.
While it's great that no single corporation owns the intellectual property behind the technology that is driving most major websites today, there is a huge price we all pay for it (whether you realize it or not). I believe that an open web is a better web, but that doesn't change the fact that it's a giant pain in the ass first getting everyone to agree on the standards and second getting them to actually implement the standards as they are written.
Flash never suffered from these headaches because it was developed in isolation by one team and then deployed everywhere. It presented something new, and powerful, and paved the way for the future. Now, the community has acknowledged its utility by making its functionality an open standard.
The implementation of Flash and its corporate owner are on the way out now, but HTML5 is essentially the exact same thing just with less rigorously defined interfaces and mostly unfinished.