There is no serious problem as such with Flash. It has deficiencies like all software have, but the issue is that Steve jobs out of his own paranoia wanted to kill it for good and started a negative publicity campaign that was so successful that people who don't know jack about technology, leave alone Flash and HTML5 started preaching that Flash is bad and HTML5 is good. It was the start of a superstition among the people that went out of control. Also a lot of hype about HTML5 is based on myths and theoretical possibilities that never materialize in reality.
Myth: HTML5 is a Web Standard and so it is better than third party plugins like Flash.
How the heck is a web standard any better than a third party plugin. After all in both cases the content is being handled by a platform which is a browser in case of HTML5 and a run-time from a single company like in the case of Flash. A browser is a lot more complex than a small run-time and it needs to handle 10000 other things and which the developers needs to focus on. Personally I would rather have one run time implementation from a single developer than have 10 different implementations of a standard which have nothing in common.
Myth: HTML5 is a Web Standard so you can code once and run on multiple platforms
When were Web standards ever standard in any of the browsers. I haven't seen a half decent website that didn't need to have browser specific code chunks to make things work properly. HTML5 is no different. It is a standard only on paper, not in implementation.
I am currently working on a JavaScript Framework involving Web Sockets. Guess what, there is no standard Web Socket implementation that is common across the browsers. Every browser developer has their own road map and implement a different revision of the drafts, so we had to use resort to a Third Party web socket implementation and guess what, it relies on Flash when it cannot use Web Socket support in the browser. So what is the point of a standard supported by the browser if you still have to rely on third party implementations. And Yeah, nothings going to change even when it does become a standard. The fact that browser developers had to make their own implementations of it makes sure that there are differences.