Correct. If Adobe had open sourced Flash right from the beginning and provided a free dev environment it may have been ubiquitous by now instead of being a glorified video codec. But the other reason Flash applications haven't taken off is simple - nobody whose opinion matters wants them to!
Microsoft is terrified by anything that would let it's locked-in customer base easily migrate to another desktop OS. Apple doesn't care so much, but would much prefer applications be developed specifically for MacOSX (and guards the iPhone like Fort Knox). The linux desktop people are busy with other stuff and distrust Adobe. The application developers would maybe like to use Flash (or maybe not) but are hindered by insane licensing fees. The only people (apart from Adobe) who really want Flash are Google, who stand to make more money if applications are pushed out onto the web. Google are the only ones who push out Flash with their browser, and include good Flash support in their mobile OS.
Adobe really tried to get people to develop whole applications in Flash, but I could never see a compelling reason to do this. HTML works well enough for most things (even more with HTML5), anything more demanding is maybe not a good candidate for implementing as a web-based application. Where is the Flash facebook or imdb? They don't exist because they wouldn't provide anything more than what we already have. Where is the cross-platform Flash email client? Nobody cares.
I don't mean to dump on Flash too much - it serves its purpose. Even with HTML5, Flash will still be used for games, advertising, and maybe video for years to come. But it will never be the all-encompassing platform that Adobe wants it to be.