What the RNC doesn't want is a challenge to a sitting president. History since WWII has shown a sitting president challenged loses the general election or the sitting president withdraws from the primary race, then his party's nominee loses the general election.
1952 Truman was challenged, he withdrew, Eisenhower defeated Stevenson
1968 LBJ was challenged, he withdrew, Nixon defeated Humphrey
1976 Reagan challenged Ford, Ford lost to Carter
1980 Ted Kennedy challenged Carter, Reagan defeated Carter
1992 Buchanan challenged G.H.W. Bush, Bush lost to Bill Clinton
I've never read, heard of or remember either the RNC or the DNC not supporting a president of theirs two years into his first term. One of the main reasons a sitting president is challenged is that sitting president looks weak and likely to lose their reelection bid.
Truman was at 29% approval when challenged, LBJ at 35%, Ford at 45%, Carter at 33%. G.H.W. Bush at 32%. Only Ford was above 40%, he barely lost to Carter, but lose he did. All sitting presidents who won reelection had an approval rating of 52% or higher. Obama 52%, G.W. Bush 53%, Bill Clinton 58%, Reagan 61%, Nixon 62%, LBJ 70%, Eisenhower 75%, Truman 55%. Truman's 1948, LBJ 1964, Ford 1976 actually weren't seeking reelection. They were sitting presidents who finished out the previous president's term. They were actually running for election for the first time. But since they were sitting presidents, I included them where I did.