If the request is for iOS, Apple is famously anti-Flash on the basis that it is insecure. Video sites that may present Flash to a PC browser, which still work on iPad, are detecting the user agent for iOS and presenting the video as HTML5 instead. Use Mercury with the PC/Chrome user agent, and video that works in Safari just sits there.
There is a bookmark trick which uses some JavaScript to divert the video through an external re-rendering web service, I think it was mentioned on this forum somewhere, but this redirection could also be considered insecure.
I believe there is a browser for iOS with Flash support built in, but I have never tried it. It would need to send an appropriate user agent, because some sites might detect the iOS user agent and just say "sorry, can't do it".