Not only that there is no perceived need by the people of South Korea to restore its monarchy, but also there is really no one that is a good potential monarch out there.
After the Japanese annexation of Korea, the Korean imperial House of Yi became members of the Japanese imperial house (House of Yamato), and the deposed Korean Emperor were accorded the status equivalent to Japanese imperial princes. After World War II, most Japanese imperials lost their status - so did the Korean imperial family.
The last Korean Emperor, Emperor Sunjong, was infertile, so the cognatic line of the House of Yi has died away. There were brothers and nephews of Emperor Sunjong, but there is a problem - many of them have joined the Imperial Japanese Army during wartime (as did most Japanese imperial family members), and this certainly would be more or less unwelcome in Korea. In fact, Yi Un, the Crown Prince Euimin, was not even allowed to return to South Korea (presumably for his association with the Japanese military).
So there definitely wasn't a positive image associated with the House of Yi in Korea - but who else could be a good potential monarch of Korea? The Goreyo kingdom is a distant past, and descendants of the Kingdom of Goreyo's House of Wang are almost impossible to find. In addition, unlike Europe, there wasn't a hereditary nobility in Korea, so there are no high nobles who could become a potential monarch. A new royal house could certainly be proclaimed - but this wouldn't be too legitimate of a new regime.
In conclusion, even if the Koreans wanted their monarchy back, they would have a hard time finding a good candidate for the throne. And Koreans didn't want a monarchy as well (why fixed something that's not wrong) - so almost no one wants the restoration of the Korean monarchy.