A non-stop flight from the US to mainland Europe is anywhere between 7-9 (East coast) and 12-14 hours (West coast) depending on the destination. There is a timezone difference of 6 to 9 hours between the US and most of Europe. Many European airports have either night curfews or quotas at night (e.g., for CDG, between 0:30am and 5:29am). Even when the airport is open at night, some infrastructure (public transport, car rental) may be closed.

Combining all of this, this means that to arrive before the curfew / night quota slots, a West coast flight would need to depart before 5am or even earlier, and an East coast flight before 11am or even earlier depending on the European airport.

This is not impossible, but it seriously constrains the possibilities.