It happens, but it’s extremely rare that hard acceleration is the best choice for avoiding a crash.
I will say yes, because people tend to treat them so casually. In US states that have motorcycle helmet laws, they often exempt scooter riders. You also don’t need anything more than a basic driver license to operate a scooter. This puts a lot of people on scooters who have had absolutely zero training in how to countersteer, swerve, or perform emergency braking. This matters because scooters can go a lot faster than bicycles; if you can’t maneuver well, then 40MPH (on a scooter) is a lot more dangerous than 15 MPH (on a bicycle).
A small motorcycle legally requires a person to demonstrate at least a modicum of agility in order to obtain a cycle endorsement on their license - although many riders eschew this step, and they are overrepresented in the crash statistics.
Many people treat scooters and small motorcycles as nothing more than cheap/basic transportation. They use it to get from point A to point B, and that’s it; they have no interest in developing their skills…and so the remain unskilled until they get into a crash.
Very small motorcycles (e.g. Kawasaki Ninja 250R) may be somewhat more forgiving than very high-powered bikes, but when you get much beyond that, the displacement starts to matter less: many of them will flip you flat on your back if you’re foolish enough to crack the throttle wide open in first gear. I think GreasyJack has summed it up pretty well: a small bike relieves you of the risk of stupid throttle-based, rider-caused crashes that comes with bigger bikes. That said, the risk to small bikes/scooters from other traffic doing stupid things is still the same as for bigger bikes.
I’ve been riding accident-free for thirty-six years and there no way I’d ever use a scooter except on country roads.
It’s good, as has been mentioned, to have an engine that is going to move you quickly. Especially riding in town.
You have to be a little fatalistic to ride on two wheels. Or certainly pragmatic. You can be the safest rider on the road but it’s always the other guy you have to watch out for and realistically that vehicle doesn’t care what you are riding.
that scoots have smaller wheels that are more likely to react badly to poor road surfaces,
they are at the lower end of the 2 wheel market - so the suspension isn’t all that good
they are light, and can be affected by crosswinds a lot more and a big bike - and in cities where you cross intersections this can be disastrous
the brakes are not all that good, try a big bike in town at 35 mph, those brakes will be super, after all they are designed to be effective at far higher speeds
there are lots of pretty inexperienced riders on scoots, of all sorts of ages and backgrounds
they are often not all that fast to hold their place in the traffic stream and so get bullied around a lot,
they don’t have the power to pull out of a closing gap - caused by an inattentive cager
the big bikes main issue really are visibility to other road users and inappropriate use of speed - but these problems are also an issue with scoots too
If you were concerned about safety, you’d go for a mid range commuter machine such as a 500 to 650 - not too fast, but fast enough.