So, let's assume that HHH managed to win the elections of '68. Perhaps the Democrat Convention is less chaotic or Nixon get's involved in a scandal which is heavy enough to ruin his chances to become President.
Hubert Humphrey and Edmund Muskie win.
I assume that relations with the PRC will remain very poor and a withdrawal from Vietnam still happens, only this time, the South Vietnamese are actually informed that the US is withdrawing. Van Thieu probably makes a ruckus, accusing the US of selling him out. I could see a US-supported coup forcing Van Thieu out of office and replacing him with Duong Van Minh(?).
Taiwan probably keeps it's Security Council seat in the UN.
I assume that HHH loses against Ronald Reagan in 1972. But who would be Reagan's running mate? Gerald Ford? Richard Schweiker?
What kind of people would serve in HHH's Cabinet?