Getting PhD from average institution (gain experience) before applying for a PhD in top institution?
Terrible idea, at least as far as a life goal is concerned. You will spend 5 years of your life doing something you consider only preparatory, without any guarantee that you will actually be admitted by the MITs and Stanfords of your aspiration. In fact, if we had someone applying to our graduate program who already has a PhD, we'd seriously be asking ourselves why this person wants to get another PhD? So it may not actually be any easier to get into the good institutions if you already have a PhD. Finally, even if you do, you will be 5 years older than everyone else when you graduate, and this will count against you (statistically speaking) when you look for jobs beyond that.

The thing you have come to realize is that everyone wants to go to MIT and Stanford, but only few actually manage to. Make your peace with this -- shoot for the best place you can to get a PhD and then make your career from there. You'll find that with a PhD from a good place in Europe, you will have sufficiently many options.