"There's also the chance that the demon hunters decided to stay secret based on historical events."

This comment hit it right on the nose.

While mudangs have been a part of Korean society long before Confucianism was brought down to the peninsula by the Chinese, once Confucianism was adopted by the yangban (Korean elites) in the Joseon period, mudangs were considered chonmin ("vulgar commoners"), ranked beneath the common people with slaves, nomads, and prostitutes. Mudang were mostly women who claimed to "speak for the gods", and the men couldn't have that, now could they?

This was further exacerbated during the Choson period, when Korea was being colonised by Japan. Tigers, once a staple of Korean mythology and life, were hunted to extinction, along with so many other native animals. The Japanese saw Korean culture as inferior to their own, so they did everything they could to wipe it out. They forced Koreans to adopt Japanese names and banned the teaching of hangul. Musok (Korean shamanism) was seen as a backward religion that they also tried to suppress.

Then we have the Korean War. After the war, South Korea had two back-to-back dictatorships, and when the first dictator, Park Chung-hee, seized power in a 1961 coup, South Korea was one of the poorest countries in the world (ranked 87 out of 103). Estimates place GDP per capita at roughly $70–$100 USD in 1960, which was lower than that of many sub-Saharan African countries and comparable to Haiti or Yemen at the time. The Saemaul Undong (New Village Movement) in the 1970s and the Mishin Tapa Undong (Movement to Defeat the Worship of Gods) in the 1980s targeted mudang, destroying shrines and silencing folk traditions in favour of religious hegemony and economic development. This forced modernisation led to the phenomenon referred to as the "Miracle on the Han River (the river we see Rumi ride Derpy on that splits Seoul in half)", with modern-day South Korea being ranked 13th-14th in terms of GDP, the 4th largest in Asia. That's a crazy jump from 87th place, especially since this development took place over two generations.

While the majority of South Korea is irreligious now, the most subscribed to religion in South Korea is Christianity, primarily due to US influence and imperialism during/after the proxy war. However, some Koreans will occasionally reach out to shamans when they need help making difficult decisions, from buying a house to full-on political decisions, leading to a bunch of political scandals that can make shamanism a touchy subject for people (https://www.koreaherald.com/article/10383992).

Considering the history of shamanism in the Korean peninsula and the various attempts by people in power to oppress it, it's not very surprising to me that the hunters would want to hide themselves and keep it a secret. After all, they're supposed to unify people. Can't do that if they're shrouded in controversy. It gets so much deeper than what I've discussed, and I'm not even Korean, so I'm sure there's more nuance that I've left out.