Hangul becoming accepted as a proud cultural heritage is somewhat a recent development. For most of its history it was viewed inferior form or writing, the superior being Chinese characters. “Hangul” is a rather recent name so I’ll use the original name, “Jeongeum”.
Jeongeum was published as a book on October 9, 1446, with detailed instructions on which letters make which sounds, and how they should be combined to form syllables. It is unclear how thoroughly this book reached the masses at the time, as very few copies of this book have been found. As you probably already know, Jeongeum was designed purposefully to give poor peasants a way to write, as the poor would not have the time or resources to learn Chinese characters. Predictably, the aristocrats did not like this notion at all, but, they dared not contradict King Sejong. Once Sejong died, everything was fair game. Eventually kings came to agree that peasants having access to easy writing was dangerous, culminating in King Yeonsan’s order (July 20, 1504) to destroy anything written in Jeongeum and punish anyone using it. This is probably why so few of the original Jeongeum copies exist.
What kept Jeongeum alive during this “dark period” was the court’s female servants. Women’s use of the alphabet was not cracked down as rigorously, because men at this time, especially 16-17th centuries, thought that what women wrote hardly mattered. Women could not take the civil exams, so had very little incentive to learn Chinese characters. However, the Korean alphabet, significantly easier to learn, was a useful way for female servants at the palace to write to family living elsewhere. Plenty of these letters have been preserved to present day — modern fonts have been designed after them, in fact — but it is unclear how far or how fast the alphabet reached the masses outside these families. It was only in 2016 that we discovered that SLAVES were using the alphabet by mid-18th century. But the general consensus is that it would have spread from female servants at court, to their families living outside the palace, and then slowly to their immediate neighbors, and so on.
Interestingly, what pushed this “lowly” unofficial peasant alphabet into furious acceptance was Japanese colonial rule. In 1910, the newly modernized Japan annexed Korea as a colony, which was immediately met with resistance by the Korean populace. The Japanese colonists reacted to the Korean resistance with various levels of brutality and/or hands-off approach, according to the political mood of Japanese mainland. By the 1930s Japan was a totalitarian military regime, and Korea, as a part of Japan, was expected to fall in line for Japan’s ambitions in Asia and Pacific; to put an end to Korea’s independence movements, Japanese colonists tried to “convert” Koreans into Japanese. People’s last names were to be changed, for example, from Kim to Kaneda, Lee to Kinoshita, and so on. In 1942 the colonists arrested a group of academics studying Korean alphabet, for “thoughtcrime” — which all but ensured that ordinary Koreans, who by this point were boiling with anti-Japan sentiment, take this alphabet to heart as a fierce symbol of Korean identity.
So it was a matter of course that following Korean independence in 1945 (and division), both South and North Korea adopted the Korean alphabet as official script, with unquestioned unanimous support by the populace. The original publication date of Jeogeum — now called “Hangul” or “Chosongul” depending on which side of the division line you’re on — became national holiday in both North and South. This is the situation that continues to this day.