Not that your logic is flawed but the situation is a bit complicated and the Spurs don't hold all the aces. Like you said, they owe it to TD to take advantage, but letting Manu simply expire will not help the Spurs at all, because they'll have only 1 swingman, RJ, under contract and will only have the MLE to use. They also have to take care of Splitter. So how do they sign Splitter and at least 3 more swingmen, having only the MLE to spend and still be a contender?
Maybe he's pressuring the FO to extend him as soon as possible, and maybe they're concerned about the money, maybe they're trying to lowball him to take a bit less, I don't know, but one thing is certain - simply letting him expire does not help the Spurs on the basketball court, only financially. And if they've given up on resigning him, then they HAVE TO trade him by the deadline.