Why would they even need Remo

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Uhhh south Korea has always been our ally.

Read a book.

What happened to the other successors? He did have them, right? I am not very familiar with the Destroyer novels, though I would very much like to get started reading them! Maybe via e-book... Hmmm.

I think though otherwise it might be too easy, and he'd find it a bore (so would we). So, that'd be no fun. But he was also being realistic and not having something of a death wish I suppose.

There's a slogan written here. 'Happiness Will Walk Away'...

I have not seen the movie, but am a big fan of the books. Based on the books, I can attempt to answer the OP's question, but allow me to explain a rough background of the books first:

Sinanju is a village in the current North Korea which has become famous for producing assassins for hundreds (thousands?) of years. The martial art of Sinanju is the root of all other martial arts, such as ninjitsu, karate, ju jitsu, etc. Very few people know this, only the really top practitioners of these arts might have heard rumours to the effect. Sinanju focuses on the art of assassins, and teaches mastery of weapons as an aside, as well as other skills such as disguise, seduction, diet, movement, art, etc. A typical Sinanju exercise is to run over wet toilet paper without causing it to crease. A master of Sinanju does not appear to be dangerous, and Remo is in fact described to be built much like a tennis player, only with thicker wrists.

President Kennedy formed the organisation called CURE as it became known that the US Constitution needs help to combat various criminal and evil doers as the justice system tends to get bogged down. This organisation is funded secretly, and has huge infrastructure in the form of supercomputers scanning all sorts of things and uninformed minions beavering away. CURE is headed by Harold Smith, a completely honourable man without any imagination or humour who often is described as being "lemony". The organisation is extremely secret and "does not exist" in order for it to disappear should knowledge of it's existence ever become a threat, as the weakness in the constitution cannot be acknowledged. Part of the disappearance of CURE includes the suicide of Smith.

Chuin is the current master of Sinanju, and his duty is to hire his services out in order to support the village of Sinanju. He only accepts payment in gold, and from the highest bidder which in today's society probably would be the US Government. There are possible political overtones should it become known that the US Government is sponsoring a village in N.Korea, as well as admitting to the deficiencies of the US Constitution.

CURE is tasked to take out any person above the law, which the organisation's supercomputers analyse to be a problem and Harold Smith identifies. Thus, in order to keep the executive arm as small as possible, only one person is tasked with assassinating the bad guys. Yes, Chuin is more than capable of doing this. However, it seems to be apparent to him that the fact that the organisation needs to be able to disappear includes the need for him to die (thus leaving his village without a source of income). So he styles himself as an ignoramus who does not understand the entire concept and can therefore be left alive. Remo Williams is required, because he is thought to be the only person who can learn the skills required as a perfect assassin as well as presumably having the drive to fulfil the duties in an understanding patriotic spirit, which means that he cannot be left alive should the organisation need to fold because he understands why the organisation needed to exist. In this regard, Chuin's final instruction is to kill his protégé, Remo Williams.

So, Chuin is tasked to teach Remo the skills of Sinanju and kill Remo, should that become necessary once CURE has to disappear. Of course, Chuin knows exactly what is going on but as part of his disguise plays along. Initially, his relationship with Remo is one of master/student, but eventually becomes one of father/son and an acknowledgement of Remo as a reincarnation of Shiva, the Destroyer. This means of course that Chuin may have a problem or two with his final instruction, which is part of the ongoing joke.

Again, I have not seen the movie, but I presume this is what the movie should have portrayed. Am I correct?

Actually, there were 2 members, Smith (information) and McCleary (out of office leg work). They found they needed an "enforcement arm" to be able to really achieve what CURE was created for. Chiun was hired by McCleary (because he had run into him during his time with the CIA) to train Remo in martial arts so that they would have their own permanent expert. Chiun was originally planning on teaching him a few "tricks", take their money and go home.