The referenced article is not accurate, as freedom of religion in South Korea is provided for in the South Korean constitution.
Article 20 [Religion, Church]
(1) All citizens enjoy the freedom of religion.
(2) No state religion may be recognized, and church and state are to be separated.
As Wikipedia notes, there was a movement to destroy the worship of gods, misin tapa undong, however:
Since the 1980s the traditional religion and shamans have experienced a modest revival in South Korea in the name of national sentiment. Since the 1990s shamans started to be regarded as "bearers of culture". Only in the most recent times the traditional religion of Korea has come to be recognised as a legitimate "religion" (Mugyo, Muism, the "mu-religion), and the notion of shamanism as the original religion of the Koreans is now pervasive.
Korean shamanism, also known as Muism, is the religion which has the longest history in Korea. Muism has always been a highly local religion, and Jeju Shamanism is distinct from shamanism on mainland South Korea. In fact, Muism is now more active on Jeju Island than anywhere else in South Korea.
Thus, a visit to Jeju Island, a center of Muism, offers the opportunity to understand and experience its practices. You may want to time your visit to coincide with the upcoming Culture Summit on Jeju during 10-13 May 2017.
Shamanistic rituals, known as gut, were traditionally performed at village dang or shrines but the more important gut now take place in more public places, both to introduce them to more Jeju residents and to invite tourists and visitors to join the ceremony.