That first video by Canberk Ulaş is a revelation. Wow. The music is as good as it gets. Albioni's music is sensational, and Canberk Ulaş plays with both artistic and technical mastery. The wind player in me freaks out seeing his cheeks blown out to cartoony proportions, which as I understand may be considered good form, but still.
Not op, but my main recommendation would be to visit greek in spring or autumn and not in summer, unless you really like it hot. Spring can be more wet, so late autumn, when most of the tourists are gone, but while it is still warm enough, would be my recommendation.
And Daemonia Nymphe for example would a greek band, that tries to reproduce the spirit of ancient greek music (not 1:1 reconstruction, but with authentic instruments).
The performances are great, the venue (a courtyard of the Venetian castle) is cool, and the host is warm and welcoming — we had a really good experience.
Also, in Athens, outside the Roman Forum site, the Museum of Greek Folk Musical Instruments is really great, sometimes has performances, and the restaurants on that street are frequented by bouzouki players as well.
If you go to an island like Paros and venture into small towns away from the tourist centers like Parikia and Naoussa, you will not only walk on ancient Byzantine roads but stumble upon a lot of traditional gatherings. Not quite platonic or Aristotelian, but still different enough from most western musical geometry.