There were private outfits here that tried a similar thing, but I don’t think it took off too successfully. These weren’t taxi companies though. You needed to ring them and book before you went out drinking. At the appointed time, a guy would turn up on the tiniest of petrol-powered fold-up scooters. He’d fold it away in the boot of your car, drive you home in your own car, then take his scooter out and ride off into the sunset.
Many clubs and pubs here also have courtesy minibuses which are either free, or charge a couple of dollars to take you home. Some do a set route, and others will take you to your door. My local club also runs a designated driver programme in which one person of a group will identify themselves as the designated driver at the start of the evening, and will receive free tea, coffee, or soft drink all night.
There are anecdotal stories (which I believe) of the NSW Highway Patrol - who have the right to set up a roadside breathalyser unit on any public road - tacitly or otherwise having arrangements with local pubs and clubs that they won’t set up a unit close to the venue, in exchange for which the venue must give the cops an easy life by not letting people get too drunk, ensuring their patrons leave quietly at night, not letting minors in, not letting a drug culture flourish, etc.
An episode of Dave Attell’s Insomniac (a show in which he goes out drinking in whatever city he has a stand-up show in and doing “crazy” stuff) featured a company called Scooterman in London. It was a very similar idea, but I don’t think you had to call before you went out drinking.
I don’t know if it would work well in the States because so many urban areas are so spread out. Plus I’m sure insurance costs (both for riding the foldable scooters and whatever vehicles your customers may have, which could be rather spendy) would probably be prohibitive.