You don't need all the AT by 15 minutes, you just need to be able to notice at 15 minutes if there's no OKW armor on the field, then in 3 minutes, OKW will probably field a KT. So you have 3 minutes to grab a Jackson and an AT gun or two. Assuming an average of 30 fuel income, 18 minutes will grant you 540 fuel. That's more than enough fuel for an American player to get a Stuart, Sherman, and Jackson out. On the other hand, that leaves the OKW player enough to get a KT and nothing else. Close to getting a Luchs, but still not enough. Jackson is all you need to force a KT to back down, and the Sherman/Stuart should be able to force the situation to where you have better map control.
You are not incorrect in your calculations, but look closely at what I said. An OKW player should recognize that you are building up AT for his coming supertank. At that point they retain the initiative because they can still choose not to build it and can divert fuel to other units. If they do, all that Allied AT is going to waste because OKW will presumably build counters to it with that fuel (For instance I see you have 3 tanks, I will get 2 JPIV, or I see you have 3 AT guns I will get 2 Stukas). On the other hand if the OKW calls in the KT they can try to force the hand of the Allied player by making him commit their AT units, and pick them off on his own terms. Basically the KT can decide where to fight the battle, stacking the odds in its favor.
The issue is not even getting AT, its that it lets the OKW player hold all the cards when it arrives. At this point they have to make a mistake, even when the Allied player has the right mix of units, for the Allied player to get in the game. This should not be, both players should have to be making plays to compensate for weaknesses in their current strategy.