What would be wrong from the Recon/Revivalist standpoint is to not also look at the background CULTURE that these Gods came from and LEARN about it if you are not of that culture to begin with (as my Wiccan example is). I come from a Greek background on my father’s side and know that I am fortunate in that aspect as it was the Greek Gods who have been the ones that called to me. And it would be very bad to say that the Gods of the Gauls or even the Gods of the First Nations of Canada (as my mother was of French Canadian extraction) could not have been the ones who chose me. I know of people who are of Hispanic, Black and even Irish backgrounds who have been called by my Gods. As a mortal, it would be hubris in the extreme to say that these people could not respectfully worship the Hellenic Gods just as it would be hubris to say that IF a god from one of those backgrounds had called me that I could not have worshiped them.
I live in a multi cultural Pagan home where Frigga is worshiped alongside of Hestia. This is because the other people in the home were called by the Gods of the Norse. The Gods do not seem to mind, because there is an acknowledgement in my home that THEY ARE GREATER than any pettiness over race or ethnicity. If Apollo, for example, can be a god of the Hyperboreans (northerners) as well as the Ethiopians according to the myths then WE should be able to include people who value the base culture in which the Gods were worshiped. We need to look beyond the pettiness of ethnic background and start to look at the willingness to honor the cultures behind the Gods and learn about them. If someone is willing to LEARN about the Greek culture so that they have a clearer understanding of the Hellenic Gods then I say more power to them.
After all, the Greeks of the Classical and Hellenistic periods DID allow those who were not ethnically Greek, but had taken the trouble to learn the language the right to participate in the Mysteries. And if the Gods were not offended then, there is no reason that they would be offended now.