Why is this a problem? Most people would and should use Chrome. Just use Chrome. Set it as default if you haven't already. There is no advantage to using the stock browser. It's mostly there as a neutral browsing solution.
So now which is it? You can't play certain mp4s or you can't download them? You've already solved (or rather, figured out) the latter....
You can always go pick up an iPhone and see how much fun you have downloading through Safari.