I tried to put it on bar 6 and then hide bar 6 and indeed it shows up again 
However, this worked a charm… I put it on bar 3 (which is active but not shown unless you press CTRL-Up_Arrow or CTRL-Dn_Arrow and that hides it from the currently shown bars and it doesn’t reappear as a floating extra action button.
Tbh I would still recommend it. You can modify it so it doesn’t go super fast as well if that is the issue.
I do as well. My bars across all of my toons are almost identical. Minor differences based on themes of mounts and professions and toys.
But I would throw the drive mount in a random corner you don’t really use. Other changes would hide that button and cause more problems later on.
This is an inaccurate statement on Wowhead’s part (so shame on them). They are not disabling ALL addons, they are removing real-time combat info from the API that SOME addons (Details, DBM, WeakAuras) depend on to provide all their capability.
Things to customize the UI like Bartender may get broken like Dominoes, but that’s not by intention.
Oh how quickly this turned into an addon removal fearmongering post.
We can’t go one day or even a quiet little post 
correct only SOME bars will hide it when its on it, believe its 1-4 not 100% sure just knew bar 3 worked so its what i tell people to use