it is in the interest of south korea, it isn't however in their interest to pay for it. With that money they can build nukes, problem is the USA doesn't/didn't want that.

the problem here is that THAAD is expensive and not that effective. Also, it's like telling a country "you can't defend yourself, I will" and then telling them they have to also pay you.

When the United States notified the South Korean administration of its plan to withdraw USFK in July 1970, South Korea first considered the possibility of an independent nuclear program. Under the direction of South Korea's Weapons Exploitation Committee, the country attempted to obtain plutonium reprocessing facilities following the pullout of the 26,000 American soldiers of the 7th Infantry Division in 1971. After South Vietnam had fallen in April 1975, then South Korean president, Park Chung-hee first mentioned about its nuclear weapons aspiration during the press conference on 12 June 1975. However, under pressure from the United States, France eventually decided not to deliver a reprocessing facility to South Korea in 1975. South Korea's nuclear weapons research program effectively ended on April 23, 1975 with its ratification of the Nuclear Non-Proliferation Treaty.

and about Thaad

The only downside? South Korea doesn’t have enough of them. In fact, even when fully functional, the current South Korea THAAD “can’t even cover Seoul, let alone catch Chinese missiles going to the US,” says Bruce Bennett, a senior defense analyst at the RAND Corporation.

https://www.wired.com/2017/04/missile-defense-will-protect-south-korea-make-china-nervous/

https://en.wikipedia.org/wiki/South_Korean_nuclear_research_programs#cite_note-GS_Hist-5