YYA> As soon as we have determined that the person putting on Tefillin is in fact a ?? ????? and has at least minimal intention to do a Mitzvah
Is the concern that the person does not care what he is doing or whether he is Jewish?
In terms of not caring, this might happen but how do we verify. Modern people are very fickle. I’ve accompanied a Chabadnik to blow a shofar to a pizza guy who continued making pizza while the shofar was sounding. I think he was paying attention, but it was really hard to say. This was not his first year, so there was no explaining happening. In the hospital, patients were aware and grateful, except I did not see those in worst condition, as he went there by himself.
As to non-Jewish: I understand that generally if someone presents himself in a shul as a Jew, we do not question his Jewishness. In theory, he might be a reform convert or even a curious Buddhist.. Can we stretch this to someone coming up to a tefilin stand that the person understands that this is a Jewish thing.