I'll try and go through this as best as I can.
In the 19th century, the rise of nationalism led to the birth of modern-day nation-states as they are in Europe, as Serbia, Greece, Bulgaria and Romania all became independent countries (and Albania later on) after almost five centuries of Turkish domination. At that time, despite the French Revolution, most nations of Europe were still monarchies- only Switzerland and San Marino were republics (this is between the rise of Napoleon and the end of the Second French Empire). So the decision was made with assent of the great powers to place members of existing European royal houses on newly-created thrones of Greece, and later Bulgaria and Romania. But the trend of importing royal houses was hardly unique at that time- after all of Europe's royal houses came from another country and assimilated/localised themselves- the present-day British and Norwegian royals being good examples.
Serbia, Montenegro, Albania and Georgia are different in that the royal houses of those countries are exclusively home-grown, and the founder of the House of Karadjordjevic was also a leader of Serbia's independence struggle, while the other three also had long traditions as royal (or in Albania's case, aristocratic) families.
The notion of monarchy being a concept alien to Greece is an interesting one for those schooled in Greek history from antiquity to the present day. Greece is indeed the birthplace of many of today's democratic ideals, which also includes constitutional monarchy. Sparta could be considered one of the first examples of such, since the kings' power was not absolute and there was a council and assembly. And other Greek kingdoms (including Macedon) had similar institutions. Then you had the Byzantine Empire, which was effectively a continuation of the Roman Empire, a continuation of Greek and Roman culture and traditions. Between then and Greek independence, only the Mani peninsula maintained autonomy due to its ruggedness and isolation.
For a monarchy to succeed it must establish a bond, a connection with the people and identify with the people of their land. It is true that the Romanian and Bulgarian royal houses, while of foreign (German) origin, have maintained that to a degree today and some sentiment in favour of restoration exists, although not quite as strong as it is in Serbia or Georgia. Perhaps, too, the standing of those royal houses might even have been strengthened by the misfortunes those countries experienced after losing their monarchies- Communist oppression, and the subsequent losses for Serbia and Georgia leaving many people looking for a rallying point (as I've iterated in another thread).
Greeks' feelings towards the Royals vary greatly- and that's partly the reason why a restoration is extraordinarily unlikely, because whereas monarchies are supposed to be unifying forces, in Greece it was for a long time a divisive issue. Many do indeed resent what they saw as political interference from the Kings and from Queen Frederica, at a time when almost all other surviving European monarchies had long become figureheads who did not interfere in politics. Paul was not wholly unpopular, but Frederica's unpopularity undermined him as it did Constantine.
Constantine ascended the throne at the time of a political crisis in Greece, and his handling of such coupled with the above-mentioned perception of Frederica sealed the monarchy's fate. In a sense, this was a misfortune because he had the chance to become a good king and one who could win wider support for the monarchy from the people- one thing I've heard said was that Constantine was "the most Greek" of all kings to have reigned in modern Greece. It is fair to say that the situation was not entirely of his own making, but little he did at the time helped the position of the monarchy.
AFAIK, at the time of the referendum, Constantine had promised that if he had been allowed to return as King, he would stay out of politics and Frederica would not return. Ironically, the very forces that had backed the monarchy up to the 1967 coup- the conservatives, church and military- were all instrumental in its demise.
Not all members of the family are viewed in a negative light- Irene and Sofia, from what I gather, do seem to be more liked and respected by Greeks even today, and even Anna-Maria seems to be such. But none of those were involved in any way with the political mess that was happening at the time and we can see that Sofia went on to become an excellent Queen for Spain, and Juan Carlos proved to be an exemplary and reconciling monarch- and had Constantine been the same, the monarchy might have survived.
Did the Royals (or at least the Kings and Queen Frederica) interfere in politics? Yes. Were there some hideous occurrences in Greece at the time? Definitely. But were they to blame for all of Greece's woes? I don't think so. If anything, it can be argued they had been made scapegoats for Greece's problems. And perhaps a further irony is that in light of Greece's present-day problems, one might have to reassess certain aspects of history- the fact that the political groups that came out of this period of Greek history must also shoulder responsibility for its current problems.
Hopefully this covers enough ground. I am new to this place, and am well aware of the forum's stance on political posting, so I've tried to skirt that as best as I can.