Why do double hung windows have sash stops

These windows must have been installed into window casements that were built in place or reused from previous windows. The reason I comment about that is that if the windows were new units with the windows themselves pre-hung in the casements it is hard to believe that they would have the misfit that you describe as delivered from the factory.

In any case you would do well to see if you can talk one or two other contractors to come out to look at the installation to see what feedback they would give to you.

Sounds to me like you need to argue the case for a replacement set of new windows. If by chance you have not paid for the job you should withhold such payment until the problem is corrected to your satisfaction. If you have already paid then you have lost a certain amount of leverage but threat of a civil suit may be in order if you do not get a suitable response from the contractor. Of course you have to weigh the cost of getting involved in litigation against the size of investment already made in the windows.

I think this probably has more to do with regional building habits than any systematic optimal decision. I agree, what you call "tilt and turn" windows seem to be more common in Europe. In the US (and elsewhere?) a similar style is a "casement" window, although it's not as functional (doesn't open as far and doesn't offer multiple ways of opening).

I have seen "tilt and turn" windows used in the US in basements, since they can satisfy emergency egress requirements. Perhaps they are less common in general use because they are more complicated / expensive? The latching and hinging mechanisms are much more complicated than a sash window, which typically is just held in place by friction.