I think thats a bit of a oversimplification.
It is correct that ROK Presidents Approval Rating has always started out very strong and ended up very low.
However this is in many cases due to the fact that every Korean President in the democratic era has had a major corruption scandal during their term. The mix of very corrupt (chaebol) politics and a strong public prosecution in S. Korea is the main cause, not the one term limit. In other countries with single-Term limits (
Vincente Fox in MX,
Ramos and B. Aquino in the Philippines, for instance) there have been many presidents with very stable popularity throughout their term.
It is also correct that if a president cannot run for reelection, (and South Koreas presidents are usually "lame ducks" for much (except the first say 1-2 years) of their Term), that their ability to implement their policy agenda is severely reduced. However the fact that this is the case in south korea is mainly due to two other reasons:
(1) A quorum Bill passed in 2012 that prevents the national Assembly from passing controversial legislation without the Agreement of at least a 3/5 Supermajority of Lawmakers.
(2) The fact that National Assembly elections and Presidential elections are not synchronised, which means that with the exception of 2008 most National Assembly elections were midterm elections, and that a President can rarely rely on a Parliamentary majority.
Koreas Presidents are already extremely powerful. They Appoint 7 of the 9 Members of the Constitutional Court (3 directly, 4 indirectly). They can veto legislation, appoint the Premier (who cannot be voted out by the National asssembly), call referenda, declare martial law, and can only be impeached by 2/3 of both the NA and the Counstitutional Court). The five-year single term limit is one of the few real barriers to dictatorship. Repealing it would make it a truly a imperial presidency, which is probably why Roh, Lee, Park and Moon have all proposed it. Those that point at Taiwan as a positive example for reelection forget that Taiwans president is far less powerful, and that Taiwanese presidents have all experienced the exact same plummeting in approval in their last term as koreas presidents have.
