I think the U.S. basically won by the Soviets' "forfeiting". If USSR landed on the moon second but soon after, say, landed a man on Mars, and the U.S. unable to accomplish the feat, we would say the Soviets won.
The Soviet program was doomed by technology and circumstance. The USSR was far behind on hydrogen, computers, and control systems; then the linchpin of their program, Sergei Korolev died (likely due in part to his term in the Gulags). They didn't give up, they failed (as exemplified by the N1 rocket failures).
No, the Sovient program was doomed because the military lost any interest in funding Korolev's trips to Moon after they got what they wanted from the program - a fully working ICBM capable of reaching USA.
After that the funding was heavily curtailed and redirected to other near-military projects like spy satellites.
The fact that the program itself was ruinously expensive for USSR (which was still less developed than US) and was burning precious R&D resources was also not insignificant.
I think there's a great argument to be made for Kruschev's downfall being the 'but for' cause of the abandonment of the program, but Korolev and technologies were the 'sine qua nons' of a moonshot.
