As a user, I prefer for my OS to do things my OS does, including styling my widgets. My scrollbars, my scrolling behavior, window chrome, all of it… whether it’s good or bad or right or wrong, that’s between me and my OS.
As a user, I hate when web apps mess with scrolling. It breaks affordances I use, like tapping at the top of the screen to jump to the top of the page, or home/end keys on desktop. It is inevitably a worse experience than native scrolling, no matter how hard you try. Facebook mobile web for example is basically unusable right now because it’s trying to overthink scrolling. If Facebook can’t get it right, I’m not sure what makes anyone else think they can.
And why would anyone want to? Honestly! Microsoft and Apple have already done all that work.
As for scroll bars and other widgets, again, right or wrong, that’s not for some web site to mess with. Sorry if the developer doesn’t like the way it looks… they only have to deal with it while working on it. As a user, I am stuck with the developer’s choices, even if they override my own choices, which I’ve made in my OS, which is where those choices belong.
If you’re going to clobber native experience, I would argue, don’t do it unless there is absolutely no other option. Corollary: there is always another option.