What way did the future discover Sator?
My guess is they first discover messages from a past agent and link these messages to the final explosion at Stalsk 12. They don't dig it up because so long as the outcome is uncertain (Schrodinger's cat), they feel they can still affect the outcome. They send lots of gold, weapons, instructions back in a long shot hoping to cultivate a past agent. They check records and Stalsk 12 has been undisturbed since the explosion so nothing was stolen from there. So no weapons/gold will be sent after the 14th as it won't affect the outcome. Eventually, they dig it up, and think they've been scammed. They rationalise that if Sator had been genuine and the algorithm stolen before the explosion, he would have told them. But they can't find any more messages. What they didn't realise is Sator WAS kind of scamming them the whole time because he didn't want to die. Since he didn't want the future to detonate the algorithm until after he dies (because he erroneously believes his actions can affect how the world time inversion happens), he sets the trigger to his heartbeat which means he is unable to say anything more to his future handlers. The future people believe they were scammed, have no more leads on where the algorithm is, thus they give up ending the war definitively.