I rode a 125cc scooter for over a year before I got a 125cc geared bike, and rode that for about 6 months before I got an ER6f (the same thing as the Ninja 650r), and my current scooter is an SH300. There are lots of things you learn about on a smaller bike, particularly in the city: observations, road positioning, hazards. Pedestrians, cyclists and cars emerging from gaps in traffic without warning. Truckers stopping to let a pretty girl cross the road while you're going down the inside bus lane, and of course you can neither see over nor through a truck. In the city, you learn to be acutely suspicious of gaps forming and avoid riding close to stopped cars, whether parked or not.
But many of these things are either specific to big city navigation and filtering, or you would learn in a car in any case. In a typical American city, you don't have the kind of filtering you see in London, much less somewhere like Taipei, and you can take for granted that the guy considering getting a bike can drive and probably has for many years.
Furthermore, there's a whole other level of learning that only happens when you're on a big bike. A 125cc geared bike is so incredibly light - it's almost like a bicycle - that you have no worries about dropping it when stopping using poor technique - braking with the front while turning, for example. And it has so little torque you'll never notice a cold rear tire slipping out on an aggressive corner, because outside of gravel on the road or wet manhole covers, it'll never happen. Not to say it can't be fun. I had plenty of launch practice: hi-rev clutch-slipping to clutchless shifting, eking out every scrap of acceleration I could. But it's not long before the stuff you're learning isn't transferable to a big bike.
So I think it depends on how much road experience you have, where you're riding, and what kind of riding you plan on doing (cities favour smaller bikes). A modern 250 or 500 probably would make a better beginner's bike. But 500cc bikes are pretty rare these days, and while 250cc are just on the edge of enough power, I don't think you'd want to have one for more than a few months, unless you're using it as an alternative to a scooter in the city.
As a bonus, they are honestly very solid balanced bikes you could ride your whole riding career. The EX250's have notable shortcomings if you will only own one bike, for many years. I certainly advise starting on a smaller bike and upgrading, but many people obstinately refuse that course. I'm not sure if they are afraid of losing money in the process, or if it's an ego thing.
As for me, I'd recommend a tiny 250cc or even 125cc for a starter. A year or two later, you can always get a bigger one safely.