Flex does anything flash does plus a lot more, no timeline as everything is coded. Many will tell you that flex created swf's are to big and thats why flash is better. To put it bluntly this statement comes from ignorance and incompetence.
There is an overhead with flex which at very best is about 250k( a few secs on a 3rd world internet connection), this can grow but at a point you switch to runtime libraries which means the first time a flex app loads it may be quite large (upto about 1.5mb). from then on any flex app that is compiled with runtimes will have a very small footprint for the calling computer, even if you have modified the application.
Application based websites usually consist of complex programming, flash bloats very rapidly when dealing with complex websites, don't confuse second rate crappy kindergarten creations of the banner ad experts with real applications. We don't live in the day of accoustic couplers or 2400 baud modems, even basic broadband deals comfortably with a 1mb website.
Today a complex html website which has several pages is not only slow but is easily broken and has no state which usually means reloading when navigating through pages. Flex apps load once and all non-dynamic content is there no matter how much the user navigates around, this again reduces time wasted with page refreshes etc.
So if you want a user to know what the 2022 html5 web experience will be like use flex, and let the client know that in 10 years time you might get close with html5 .
You can defenitely do all the stuff you need in Flex which you can do in Flash ..Infact you can do easily the design stuff using mxml tags which you can use to structure your UI.
You also have the Flexibility and source to use the skins and animations developed in Flash to be used in Flex as well....In Flex you have very much improvised and advanced DataService techniques to interact with ServerSide data and built in support for Charting applications and much more....
Yes, as the rather enthusiastic replies say, Flex is definitely up to the job and your summary is right and mostly the replies you've received are right though flex isn't going to displace the Flash IDE anytime soon.
What hasn't been said is that while Flex will build a great functional website, the appeal will be highly dependent on skinning to get away from the workmanlike appearance of standard Flex components, so using Flash you can build/fabricate those skins, but with Flex you will need rely on other software to build the skins for flex.
The workflows for building such sitesin Flash and Flex are differrent and that would depend on your creative streak and how you approach development. It certainly isn't a done deal that Flex is better for this job and a number of people who love to develop with flash will find Flex heavy-going at first.
You won't really be in a position to make an informed decision until you get more proficient and the site you mentioned is not one to attempt as a first project - simplified, maybe.
Thanks for wounderful answers you wounderful guys