As for Greece, the migrants who end up there probably weren't looking at Greece as a final destination, but as a stepping stone for better destinations in Europe.
That's exactly one of my concerns here. Why doesn't for example Jordan just let them assimilate in their society, as fellow Arabs, and instead it just keeps them in camps? Of course not just Jordan, but also other more resourceful Arab/Muslim (mostly Sunni Muslim) countries such as UAE, Qatar, etc.
Wouldn't it be easier for these already traumatized refugees to integrate in Moroccan, Tunisian, Egyptian, Jordanian, Saudi Arabian, UAE-ian(?)...societies? Isn't there a "brotherhood" thing between Arabs (maybe even between Muslims? or just Sunni Muslims?)?
As for why the rest don't, one answer is that the other countries don't have to, so they don't give a fuck; and even more cynically, by keeping the festering sore open, they can deflect public attention from their various problems and instead blame Israel for it all.
I picked Jordan because it's just a neighbouring Arab country, plus mostly Sunni Muslim, just like Syria, and unlike Lebanon. And of course the Palestinian refugee issue is another topic entirely.