My mind has a picture of what an umbrella on a bicycle would look like. To make such a thing possible or practical to the average cyclist, I think requires a redesign of the concept. When I'am out riding, and I encounter precipitation, the question I find myself asking is, "would putting something between me and the precipitation make me more or less comfortable". Often if the rainfall is light, I will put nothing additional on with the expectation that my exertion will negate and evaporate the wetness that occurs. If the weather is hot or very warm, putting nothing on is kind of a relief. The only problem one encounters is the shock to body of cool raindrops. Once you are wet, as in totally soaked, it just becomes your reality. Putting on raingear, in such a situation could make a person sweat so much that they need a shower anyway.
It becomes a question of balance between protection from the rain and ventilation for the excess heat. I think an umbrella type system would need to be aerodynamic but not get in the way of forward vision and rider safety. I think the traditional umbrella shape, function and deploy ability would have to be entirely abandoned. That said, you do need a system that would be easily set up and taken down, all the while being flexible.
The big hole in the currently available options is the space above the neck. One can get a helmet cover that repels rain. But the big problem as I experience it is the face. I have wondered if some kind of positive pressure face mask could work to keep rain out and yet not get all fogged up with respiratory vapors.