I don't think it really matters that much. Learning on one may build necessary qualities faster than another, but other qualities will lag. Maybe we should look at accident stats during training and the first 500 hours after training. Anything that predominates there should be what is focused on in training. If flying a different type will help reduce accidents, then go that way.
In reality, what we have now is very safe, effective and fairly cost efficient. Why change it? Okay, lower the cost, make it even safer and more effective....methinks it would have happened by now if it could have.
Now if we're talking carte blanche approach to training then I'd say fly pistons, turbines and twins, everything from non-governed to FADEC, etc. Maybe a variable stability fly-by-wire twin turbine machine to simulate many different types of helicopters but can also simulate piston engine performance and indications.
______________
Where I would advocate change is in syllabus. Basic handling and aircraft knowledge seem to be the strongest focus. CRM, regulations, and decision making are things that I'd like to see improved.