There’s nothing at all weird about a “white European” becoming a Melkite priest, or a Melkite at all, for that matter. It is not part of the native historical culture of anyone except Syrians and Lebanese, however, ethnicity means little in one choosing that rite — I am not Italian, nor did any national ancestries in my family tree speak Romance languages, yet the Latin Mass is part of my very soul. (For the sake of argument, English is a mash-up of a Germanic and a Romance language.)
However, if I were a Melkite bishop, and a single young man came to me requesting a change of rite, the very first question I would have is “are you changing rites so you can first get married, then become a priest?”. Any man who is serious enough about religion even to prefer one rite over another, and to seek out a rite that isn’t common in his part of the world, is ceteris paribus more likely to discern a priestly vocation, than a man who never gives different rites of the Church a second thought. The typical Catholic might be only vaguely aware that different rites even exist — many have never heard of them, or may have heard but paid no attention, perhaps thinking “they’re kind of Orthodox, aren’t they, whatever in the world that might be”.
I would be asking “what is it about the Melkite rite? why our rite instead of another rite? — there are several, you know”. When I became a Latin Rite Catholic, no one said to me “hey, just in the interests of full disclosure, we’re not the only rite in the Catholic Church, you know, if you’ve never heard of all these other rites, you might want to check them out before you make a commitment to the Latin Rite”.