Usage of power trolleys in GB&I is huge. I would estimate that 60%-70% of men and probably 90% of women use them at the courses I normally play.
Weather conditions and the hilliness and dampness of terrain play a big part though - for example the need in GB&I to have access to heavy/bulky items like brollies and waterproofs for most of the year, even snacks and drinks too, and such things all add up in weight terms and the older the player, and let's remember we're discussing all ages of players here including older seniors of both sexes as well young fit bucks, the more the weight and bulk effect matters.
Before power trollies came along pull (never push) trolleys with not very good wheel bearings were the norm in GB&I and what a pain they were, especially on a hilly or damp course....aching arms and general fatigue etc from pulling the trolley not from the golf. I can remember having to oil the bearings by sliding off the small metal cap, just like on a bike wheel, which is what many of them were, small bike or pram wheels with spokes. Latterly wide wheels came into fashion, allegedly to prevent damage to courses.
Over the last few years the lightweight and ease of roll of push (or pull) trolleys has increased significantly, which I suggest is why they are used more these days, but seemingly rarely by older folks, seniors or the infirm and much less so on hillier and damper courses. 'Strange' how the narrow free-rolling wheels don't damage the course though (see above).
As to electric trolleys, I have used one in the UK for years. I've never had to have to second 'just in case' battery with me nor been let down and had to 'push-in' from out on the course. There is a great variety in quality and costs of power trolleys though, and as usual, the pay-peanuts-gets-monkeys line is appropriate. It's also worth mentioning that these days lightweight lithium batteries, although more costly to buy, are smaller, weigh less and charge quicker than larger size 'conventional' batteries.
There are all sorts of means of moving your clubs around the course and all have there place. There is imo no real right or wrong way, just what is appropriate to the individual, the course, the weather conditions etc.
What I don't think is appropriate however, is golf clubs imposing rules on what method players can or cannot use.
As to power trolley usage in the US, well heat/humidity in some areas at some or all times of the year may be an issue and I can see the vested interest opposition in carts-making-money for clubs and the protection of caddy programmes. I am though, very surprised at the purchase cost of power trolleys (in Europe well) and the charge to store them at the club...most folks in GB&I seem to keep them in the garage and charge them ourselves.
Atb