I can only base my comments on my own experience of 3 days/2 nights in Syros -- and I would not say that Syroites are unwelcoming to non-Greeks, it's just that, in my experience, they pretty much ignore them. If that's OK with you, ,and you have traveling companions to talk with, fine. Also, while the main Square of the port city & capital Ermopouli, is visually stunning (and the reason many people go there, just to see it), the rest of the island, such as landscape & beaches, didn't impress as much. I went there because of that sight, going solo in May after stays with friends on other Cycladic islands, and then moving on to Athens.
Furthermore the port town had very much a "business flavor" rather than an air of style or local arts/crafts/academic interests. While there I realized one could only get a picture of the town square and the background (2 major churches on hilltops) from an aerial perspective, not one's camera. Thus, I looked around for some postcards of same -- the photos seen online and in guidebooks. Guess what? I searched up &down streets of the port for nearly an hour, found NO places with postcards or souvenir photos (also found almost no galleries or shops showing local artistry or handmade objets or fashions) ... only when I got to the port, just opposite a pier for day-excursions, did I find a single rack of cards.
There are historic reasons for this: (1) From 1850s - 1980s Syros was THE place for fuelling All ships plying the Med-Aegean. Thus is was a huge business center, became capital of Cyclades, home office of many banks, realty companies, brokerages, law firms etc. This wealth also produced the mansions on hills, and in areas near port, many of which are converted today into condos or hotels. Syros to this day is VERY business-oriented, not arts-oriented. (2) No postcards or materials for tourists because apparently very few visitors to island are non-Greeks. Vast majority appear to be GReeks who have family there, or who return annually for a holidays; why would they need postcards for a place they know so well?
As I said, I was not mistreated, just ignored, in restaurants, at the beach, in shops etc. 99% of people appeared to be Greeks, and not newcomers -- all busy talking with one another. No other non-greeks around that I encountered, on the beach, on the bus, in restaurants, for a touch of sociability. I've been on more than 20 Greek islands n past 20 years, often on little solo detours like this, and have never had this experience. On those occasions, english-speaking tourists also could be rare, but I always found some friendly locals to chat a bit with. If you have friends along, your experience may well differ.