I think yours and typhoons have got it about right.
But I wouldn't personally bother with doing the FAA route unless you have a green card or have an idea that you will need an FAA license for a future job.
The N reg stuff in europe is all up in the air at the moment so you might have an advantage over some other students if you have both tickets for flying N reg Biz jets if the require you to have dual licenses.
But we shall have to see what the fall out is of EASA.
Going intergrated in the current climate is plain stupid.