Why does Doctor Strange not bind Thanos in a time loop in the same way as he did with Dormammu in the Doctor Strange movie?
Because Dormammu doesn't exist in time, but Thanos does.
In Dr. Strange, it was stated that Dormammu belongs to the Dark Dimension and existed outside of time.
Wong: Dormammu dwells in the Dark Dimension. Beyond time.
Generally, when you reverse time, you reverse everything including the memory of the person existing inside the time. Since Dormammu doesn't exist in the time, his memory does not get reset, therefore he gets tired of seeing repeated thing for many times and is defeated.
Since, Thanos exists inside the time, his memory will be reset, therefore the time loop won't make him tired and fall like it did to Dormammu. Thanos will do he is supposed to do every time and Dr. Strange will only reset his memory after the end of each loop.
Strange planned to lose.
It seems odd to me that he just handed over the time stone without a fight to save Tony. Especially after the speech he gave about how he'd let Tony or Peter die if it came down to saving them or protecting the stone. He was definitely willing to "trade lives". When asked why he offers "We're in the endgame now." Then later when he dies he explains "It was the only way".
We're led to assume that when viewing possible futures that Strange was only looking as far ahead as the upcoming battle with Thanos but given what he said after they lost it seems likely that was looking further ahead. Strange saw a future wherein in order to win in the long term they had to lose in the short term. Dr Strange threw the fight because he saw Avengers 4.
Everyone here is correct (ie the only winning future was the one where they lost that fight) but there is a more obvious reason why any time loop future is a loss one (assuming he would be able to loop Thanos as he did Dormammu in the first place).
Because Thanos knows what the stones are, and it will go for that instead of killing Strange. In a position where Strange creates a time loop, he has to use the stone to do it (ie, not be hiding it). Then Thanos has a target. Instead of killing him over and over again and resetting the loop as Dormammu did, Thanos understands the power of the time stone and that Strange just gave him an infinite number of attempts to try and get it. Eventually he will succeed in wresting the stone off strange/breaking the protective spell (if only finding the solution through brute force search), in which case he can end the loop with the stone in his possession.
Strange didn't attempt a time loop because, just as others stated, he knew it was hopeless. He had searched through millions of different outcomes, and the only one with victory, was to surrender the stone and let Thanos think he won (i.e. the smug look of satisfaction on his face at the end of the film when he's sitting at his house).
Strange knew what was going to transpire, even if it meant his death (temporary??). He knew what Fury was going to do before he faded away, he knew that attempting a time loop would give Thanos infinite number of times to gain the stone. As result, he resolved himself to surrender, and told Tony "This was the only way". It's a life lesson that sometimes, the only way to win the war, is to allow the enemy a strategic victory by strategic retreat/sacrifice. He knew ultimately, they'd win the war.
Makes me excited for Infinity War Part 2!!!! :D