I believe the idea of an interesting endgame wasn't that randomness would make the weak strong, but people would be amassing different types of power (Adventure/Magic/Diplomacy/Military divisions being the most obvious) that would clash in interesting ways instead of a "battle of the biggest". No randomness at all was mentioned in Frogboy's examples of endgame turnarounds; unexpectedness, yes, but the person planning it still worked for it and intended the outcome.
Yeah, i think the only working solution is to have different kinds of power that allow a player to win in different ways.
Like, say, in Star Chamber (i wish it wasn't a CCG, i really like game mechanics, and it's an online game, but i don't want to waste money on a CCG) there are 3 different victory conditions - Military Victory (conquering the opponent’s homeworld), Cultural Victory (having a “destiny point” advantage of 30 or more points; destiny points are generally gained by controlling “artifact planets”), and Political Victory (winning three Power Play votes). So, even if you're close to a win, your opponent may be close too, but in another category. Also, you know about how close your opponent is to a win (you have a 100% scouting of the map but you don't know enemy hand). You must chose what will be the most efficient - to use your assets to slow down your opponent's victory, to fasten yours, or both.
I played only a free tutorial, but it seems that endgame in Star Chamber is potentially the most interesting one of all turn-based TBS i played in 18 years. The essential features to such success seems to be
(1) several winning conditions and different "power" levels required to reach the winning conditions
(2) an ability to counter the opponent moves so to delay them
(3) vast knowledge about the current situation in the game, even if not 100% complete
So, (2) gives you an option to either speed up your own victory or slow down the enemy victory, whatever is more efficient. That alone makes an endgame more intelligent and less boring and straightforward. It's not just "make a 10000 clicks to stearoll everyone with your army" or "press end turn 200 times for a technological victory".
(3) is essential because it effectively enables (2). If you have no information, you can't make a strategic/tactical decision about what move will probably be the most efficient one. If you don't know your enemy, you can't counter him and you can't counter his counters.
(1) means that the game is not just a one-dimensional game of one big number (say, a number of cities you have). You can have different goals and different objectives, and you don't just play a game of "who has the most of X".