There's 5 turns to "brilliant victory" after the AI turn, but it comes up "Axis loss". What am I missing? It's happened twice. I have two of the three objectives. Am I making an error that's killing my game at this point? How can the AI win without taking any objectives?

There is a Soviet recon unit which will capture your Axis victory hex if unprotected (also see the thread linked there). Might the capture of the original Axis victory hex trigger the loss?

Capture of original victory hexes do trigger a loss. You have to defend them or be certain the enemy does not penetrate your front lines.