I'm writing from the UK perspective here.
Electrically powered trolleys are massively popular in the UK. I use one, and have done so for years even though my preference would always be to carry if I could. Without electric trolleys many older or infirm folk, especially on hilly courses, would not be playing at all or only playing a few holes. I would guestimate that at my club about 70% of male players use an electric trolley, the other 30% either using the modern push trolley or carrying. For ladies it's probably more like 95%-5% split with none at all carrying. This does of course reflect the age profile of a club, but that's another issue for another debate. At most, but not all, UK clubs, the Pro will have a few of electric trolleys to rent out, which is very useful for visitors etc and makes him some revenue.
Also popular in the UK are small one-person electrically powered buggies, sort of like high spec disabled buggies. These normally have handlebars rather than a steering wheel and no front windscreen. These machines, the use of which is normally restricted to those with some physically limitation, or of a certain age or illness or disability, are surprisingly 'grippy' even when conditions are wet. As to twin-person buggies, well these should only be for the old/infirm/disabled IMO and then not allowed when it's too wet/slippery - twin person buggies with those fat smooth types are far more prone to going sideways wet it's wet, even with just morning dew, than the single person buggies, which have thinner tyres. Another consideration with buggies in the UK is the temperature - with limited physical movement to keep you warm buggies from late-autumn to late-spring can be damn cold and staying out for 18-holes a distinctly chilling physical experience.
It is also worth pointing out that there is a physically disabled golfing element in the UK with golfers actually playing all their shots seated in their specially converted buggies, and good luck to the folk involved too, it's great that they are able to participate in a game they love.
One interesting aspect of electric trolleys, and the new generation of push carts, is wheel width. Years ago, before electric trolleys in fact, all trolleys, either push or pull, had very narrow wheels, bicycle width, some even had spoked wheels. Then in the '80's, 'someone somewhere' thought that wide wheels were less damaging to the course so narrow wheels were discouraged/banned and folk had to buy new trolleys or convert their existing ones. A nice money making venture. Now, the wheel has turned, no pun intended, and the modern generation of push or pull cart has narrow wheels. Crazy world sometimes.
All the best