If the "designer" only cares about the look of their site, then they're an artist, not a designer. Usability and accessibility are intrinsic parts of web design.
And if their brand is "confusing and hard to use" or worse, "glitchy" then I guess making their own scroll bars is appropriate.
Back when splash pages were popular I pretty much closed any tab that presented a splash page -- life is too short. Likewise whizzy Flash sites. What I miss about Flash is that it was an excellent bozo avoidance system.
Obvious solution is obvious. If the user is using Firefox, instead of replacing the scroll bars, replace the page with a message telling them to switch to a better browser. That way everybody wins and it's so much less work.
Or more obviously - if a browser works a certain way, and the user has chosen to use it, let it be. If the designer doesn't like it, well - the user chose to use this browser, and they choose to use your site. If you start trying to change one of their choices, they might change the other.