I think your concerns may be best addressed by a dual-SIM phone rather than an all-purpose SIM for both the US and EU--although Google Fi does offer that flexibility.

A dual-SIM phone also addresses this issue though and probably cheaper than Google Fi unless you're traveling to Europe more than three or four times per year. A dual-SIM does what the name implies: it can work with two different SIMs so you can use one for US service and a separate one for your travels. For instance, with my dual-SIM phone, I keep my giffgaff card in the phone even when I'm in the US while switching my service to my US carrier; when I'm traveling, my primary service becomes giffgaff, and I don't use my US service.

If you really want people in the US to be able to reach you through your US number, then you should seriously consider Google Fi.