We reimplement Scroll because browsers like FireFox don't let you apply CSS for scroll bars, forcing you to have the OS's scrollbar branding (colors, style, etc..) in the middle of your page's branding.
The last thing you want to do with your brand is to have someone else take it over.
Understood but I think you got on the wrong side of the tradeoff. 99.999% of sites use the browser scrollbar and that should be a hint (check apple.com and tell me they don't care about branding.) Do you really want to be remembered as the site with a great tech demo AND a bad scrollbar? How can that be an improvement over just "a great tech demo"?
Furthermore I don't appreciate brands taking over my desktop and reducing the usability of my browser (and your site) but that could be just me.
Seriously? You think you can do a better job at implementing scrollbar than browser makers? And you have the resources to test on every_single_device out there? And you really think I would even notice the scrollbar if it worked?
Browsers also don't let you apply your branded CSS to my address bar, or to the clock in the corner of my screen, or to the plastic framing around my monitor.
So you break scrolling for a very large number of users because it looks marginally prettier? That's, like, the literal opposite of proper software design.
Reality check: not a single user gives a damn about the pretty scrollbar. They care about being able to use the site. Breaking the latter in order to achieve the former does the precise opposite of what you intend: it weakens your brand because users associate it with broken websites.
This isn't to mention that a user might very well have a system scrollbar set for usability reasons (perhaps with a more-visible - whether better contrasted or simply bigger - slider due to poor vision, or an oversized slider due to poor motor skills), which means that your prioritization of "brand" over function is effectively giving the middle finger to people with disabilities. Nice going.
There will always be some element of the user experience you can't control. What about the shape of tabs, or the colour of the [X] to close the window? I think the scroll bar is in that category - it has nothing to do with branding and a browser scroll bar doesn't devalue any brand.
I just checked and perhaps the biggest brand in the world - Coca Cola - uses the bog-standard scrollbar. So if it doesn't ruin their brand, it is probably OK for you too.
The last thing you want to do with your brand is to have someone else take it over.