The US also directly supported this and helped many people who otherwise would have stayed. It was part of a dilibrate strategy to weaken the North.
Many of the fears people had about what the Communist government was doing were false, and the Communist government actually implemted some sensable policies.
At that point it was not clear that the North would get more support, but over the coming years it became quite clear. The South just failed as a state way more so then the North.
A bunch of poor Vietnamese were able to defeat the biggest and most powerful military in the history of earth. That's a pretty hard thing to explain.
Also, lets remember that we should not take everything they say for granted.
A lot of resistance is also passive and that is effective as well.
First of, the US did not invade North Vietnam because they were afraid of war with USSR and China. Not for any other reason.
The US absolutly dominated the South, it was a client state. They basically made it possible for the South Vietnam government to exist. When the government there did perform as the US thought it should, the US installed a new government.
The US was the only thing preventing the South from failing. You are suffering from Propaganda if you believe the South Vietnam was much of state without US support. It did not have the support of the majority of people in Vietnam and probebly not even in the South.
The government that the US supported was handing out land to political cronies against the people on the land, who did you think they were fighint? Then the American went in and helped this corrupt regmie to enforce these policies.
The US Army absolutly slaughtered the people of Vietnam in unbelivable numbers. They spread 10% of the country with chemicals that lead to 100000s of deformed children. The numbers are stunnig and I could go on about all the other countires and people the US bombed and how that helped to great the worst genocide in human history (% based).
And all of this for what? The leader of North Vietnam wanted to firendly with the US in the beginning. Even basic geopolitics could tell you that he would much rather have the US as a supporter in order not to be dominated by China. This actually what has now happened, Vietnam is now US new best buddy. Also Vietnam turned capitalist pretty quickly once the US was gone. Most people realize that Communism is a idiotic idiology and most elites don't really like it at all.
Vietnam might not be rich yet, but they are doing much better. This process could have been started 20-30 years earlier but they had to fight a war for survival.
Your simplistic comparison with Korea is just that, stylised to fit your narrative. In reality the two situation were quite different.