The real problem of flash is just a liiitle thing: it's ass to navigate.
You are rarely able to select or copy anything, the context menu is fucked up, scrolling rarely does what it should do, it's fixed layout (ever gone to a site "optimized" for 640x480?), shitty artistic fonts, clever navigation, et cetera.
I'm not against using flash, but I'm against flash sites. And if you could prove the existence of a usable flash site please do.
In short, unless you are the brother chaps, don't make a flash site.