Flex is good if you want to install as an application using Air. ebay actually has a RIA version, or at least they did for a while.
There are several concerns with a Flash site, some of which Adobe has addressed in the last couple of years.
One is accessibility. It is more difficult to make a Flash site accessible to the visually impaired, who use screenreaders.
Related, people like to be able to copy and paste content, as well as print. With standard web pages, the developer need do nothing to support this, it just works from the browser. With Flash, you'll have to take care to make your text content selectable, and I'm not sure a user could ever select a section of text + images. I've never done it, but I can only imagine that implementing printing ability in your Flash site would be nightmarish, at best.
Another is linking and addressability. For most sites, you can copy or at least get a link that you can share or save to get back to where you want to be. With Flash, you usually go to a single address and then the rest of of your experience happens inside the Flash "window" and the browser isn't actually navigating any pages. It is far more difficult to implement this ability in a Flash site.
A Flash site is usually very slow to load, especially the first time, and page load times are immeasurably important to a site's success.
Finally, you've perhaps heard the furor over the last couple of years about different vendors supporting or not supporting Flash in their products/devices. An all-flash site would be completely inaccessible on any iPhone or iPad, for example, which do not support Flash at all. Furthermore, some employers do not allow plugins like Flash to be installed on work machines, so you'd eliminate that portion of possible traffic, as well.