Things are not black or white.
In all the countries founded in the 19th century -the century of revolutions of independence-, the monarchs were appointed, and their popularity rested on how well they achieved to become 'nativized' after being installed. It is fair to say George I did a very good job in that regard -there was virtually no republicanism during his reign. His son Constantine was initially even more popular -particularly after leading the army at the Balkan wars which doubled Greek territory. For the greatest part of the Greeks, he was the personification of the king they had been asking for in the national prayers since the fall of Constantinople in the 15th century. But then the national division between the supporters of neutrality in WWI, headed by Constantine, and the supporters of entering the war in the side of the Entente, headed by prime minister Venizelos, turned a good portion of the greek public against him -although he remained hugely popular for the greatest part.
After the fateful ending of the Greco-Turkish in the aftermath of WWI, which Venizelos had began but was concluded under the leadership of his political opponents and Constantine -who had been temporarily replaced by his son Alexander during Venizelos' primeministership in the midst of WWI-, the more extreme supporters of Venizelos came into power and abolished the monarchy -although the referendum of the 1920's (fraud, to be sure, as all referendums in the first half of the 20th century) showed that, among the old inhabitants of mainland Greece (that is, excluding the refugees that fled from Asia Minor after the war ended) the popularity of the monarchy had not been significantly affected. In any case, the 2nd Hellenic Republic which lasted 11 years was a complete failure, filled was consequent coups and anarchy, and the monarchy was restored for the first time in 1935, when another referendum brought back Constantine's son, George II. The referendum was certainly fraudent as all of them were, but historians agree that a popular majority had been formed in favor of restoration (see: Richard Clogg's 'A concice history of Modern Greece').
A year later the King imposed a dictatorial regime that prepared Greece for WWII (this time in the side of the western allies), and also purged the handful of communists that then existed; overall though, the success of the regime in pushing back the Italians at the beginning of WWII ensured popularity. A little after though the Germans conquered and occupied Greece for four years, leading the legitimate government and the king to escape to the Middle East. The communist party held the banner of resistance inside the country and grew immensely popular during German occupation, leading it to succumb to the temptation of attempting to conquer power after liberation. It failed thanks to English intervention and support, but the legitimate government promised the king would return only after another referendum. This was held in 1946, and as in the meanwhile the populace had been largely scarred off by communist attrocities and the fear of a generalized civil war, the king's return was again approved. After his restoration, the last and main phase of the Greek civil war commenced -during which George II died and was succeeded by his brother Paul-, and after its ending, thanks to American contribution and aid, two decades of domestic tranquility, growth, but also anti-communist repression continued. The monarchy was identified with the anti-communist state, which caused the youth -that grew increasingly left during these decades- to become allienated and turn against it, but still the worst would have been prevented had not Constantine II, only months after succeeding to the throne, handled in a profoundly unconstructive way the political crisis of 1965, as a result of which the centrist government and its overwhelmingly popular leader George Papandreou (grandfather to the recent prime minister) were edged-out. This caused a chain of instability, which in two years-time led to the military 'intervention' of 1967 -the 'junta' that lasted 7 years. After its ending, the king was blamed by most for the turn which events had taken prior to the dictatorship, and with even the centre-right party of Constantine Karamanlis avoiding to support restoration, Greeks voted overwhelmingly against it in the referendum of 1974.
This is the story -in short- of the Greek monarchy. My conclusion was that it was a lot more popular than in its last days at several points in its history, and also had a handful of opportunities to remain in power and be assimilated more constructively in Greek political life. Unfortunately, bad luck and bad choices prevented this development. They may be blamed for a good portion of that -but not all though. Greece is certainly a tough country to rule -by any standards.