Oh, and another thing: According to Archer's bio in "In a Mirror, Darkly," he retired from Starfleet in 2169 to become ambassador to Andoria, then became a Federation councillor, then the President of the UFP. Granted, that wasn't stated in dialogue so it isn't strictly canonical, but if you accept it, then nobody is going to refer to a former president as "Admiral Archer" instead of "President Archer." So it would have to be someone else.
Plus in The Next Generation we saw an incredibly aged Leonard McCoy and it has been established that life expectancy in Star Trek is somewhat extended than what we have currently.
We saw a McCoy who was nine years younger than Archer would've had to be, and he was portrayed as exceptionally ancient. And that was with the benefit of geriatric science more than a century beyond what Archer would've had available. By analogy, it would be far more reasonable to postulate a character from a present-day series living to age 100 than it would be for a character from a Western.
Using your criteria then we don;t know for sure that it was McCoy we saw in Encounter at Farpoint. We know that the person was saw was an admiral in Starfleet. That he had a certain affection for ships named Enterprise and he was familiar with Vulcans. He's never referred to by name. He was played by DeForest Kelly but many actors have played multiple roles. At best we can say that it's possible, even likely, that it was McCoy but we don't know for sure.
BTW, Dwight Eisenhower was and is often referred to as General Eisenhower as well as President Eisenhower.He wouldn't be referred to formally as General while he was president but quite often and informally he was referred to as General after his presidential tenure was over.