And JFK also had the CIA "dispose of" diem.
To be more specific in Laos, the Hmong were given arms and training under Vang Pao, which aimed to subvert the laotian government, and thus Laos got a finely woven handbasket to hell with the rest of indochina.
Eisenhower let the CIA pull coups and assassinations in the 50's (Iran, Guatemala, etc.), and those were democratically elected governments which were overthrown simply because the US feared they'd have ties with the Soviets. Diem was a tyrant who was not so much democratically elected as denying any democratic process to question his power, and was not "disposed of" by the CIA nor Kennedy. Rather, Kennedy and the CIA
allowed a coup to take place by the military rather than funding it or carrying it out themselves believing it would lead to a more democratic and effective and competent government. And CIA operations in Laos began under Eisenhower's administration.
Regardless, if Eisenhower didn't get any flack for it nor had it revealed to the general public nor had it lead to his impeachment, there's little reason for Kennedy to.
JFK also had links with the Mafia (remember the iffy eleciton count in illinois)
Highly specious. John F. Kennedy did not have links with the mafia and if his father did is debatable and enough to declare it not a fact as you present it as.
There was no impairment up until the time he died. But had JFK lived longer, the medication could have been increased, as he had severe back pain. At some point, his cognitive abilities could have been challenged. Then, he could have faced pressure to leave office.
His back was gonna hurt the same in 1962 as 1966. It's not like he'd exponentially grow worse and worse after November 22, 1963 magically. Slowly over the decades after, perhaps, but the condition wouldn't go into hyper mode in 1963 and after for no reason. Likewise, it'd be highly debatable whether his back would grow really that worse over the years to come and I wouldn't say it would go that much more down hill to dope Kennedy up till he was tripping or a vegetable or something.
In conclusion, an impeachment is interesting and an interesting story concept, don't get me wrong, but it's highly implausible in actuality. I mean, the things they say he'd get impeached for are very unlikely to ever get revealed, unlikely to destroy him if they were ever revealed, much of the rest (such as the mafia,etc) is dubious, and the very process of impeachment is unlikely. The reason Clinton got impeached had largely to do with political revenge by the GOP for the condemnation of Reagan during Iran-Contra, and the negativity toward the GOP during that time. In this TL, there is no such thing to get revenge for so the most likely thing to get revealed, which is sex, is a hell of a lot less likely to lead to impeachment (and especially given the context of a world where sex is hush hush and every politician and prominent person is screwing someone behind their wife's back). Perhaps the GOP loss in '60 and a Republican belief that Kennedy had stolen the election, but that's still not the same thing as you had in the 80's and 90's where there were calls for
impeachment of Reagan and the GOP got revenge on Clinton, and the Republicans lacked a presence in the Congress to do anything. So there is just too much working against it.