I, unfortunately, cannot speak on the history of Taoism in Korea as I am simply not well-versed on the subject. However, I can possibly shed some light on the Japanese side of things.

This is a very interesting topic. As I am sure you're aware, Taoism was a Chinese indigenous religion. The Japanese indigenous religion, Shintoism, took influence from Taoism when the practice was transmitted to Japan in the early seventh and eighth centuries. These things included, for example, the doctrines of yin and yang, along with the five phases of cosmology (further reading listed down below). However, there were no Taoist rituals, ordained Taoists, or monasteries that made it in this transmission to Japan, which is maybe one of the reasons why it might not have spread well (this is my very possibly wrong guess as to why). A lot of these influences found their way into Japanese Shintoism, some of which are even seen today.

So, I think the answer to your question, or at least part of it, is that while Taoism isn't explicitly practiced as Taoism (if that makes sense), it influenced Shintoism, which became popular in the premodern period of Japan. This popularity was replaced with Confucianism and Buddhism, as you correctly said, during the Edo (or Tokugawa) Period from 1603-1868. These practices (I'm hesitant to call them religions; Neo-Confucianism had an emphasis on philosophical thinking and how to live life rather than being a religion), however, were replaced with State Shinto after the Meiji Restoration in 1868 (ish). After this, State Shinto in a sense became the official state religion, as the legitimacy of the Emperor came down to whether or not Amaterasu (Shinto Sun Goddess) chose him to lead Japan (pretty much because it was good for nationalism). This formation of State Shinto is kind of when Shinto started being practiced as a distinct religion, rather than just the way of life for the Japanese.

I understand this is a very surface-level answer, but I don't have the appropriate sources on me to give you more in-depth information while ensuring its accuracy. A lot of this info is knowledge I acquired in my Japanese history classes, and the reference I have below. If anyone sees any inaccuracies, please let me know!

If you want to learn more about Neo-Confucianism in Japan I highly recommend this: https://www.jstor.org/stable/2718738

References

Kohn, Livia. "Taoism in Japan: Positions and Evaluations," in Cahiers d'Extrême-Asie, p. 389-412. https://www.persee.fr/doc/asie_0766-1177_1995_num_8_1_1103