Every team has a big board. That doesn't mean you shouldn't weight in information from other teams' needs when you have multiple picks.
Say we get to 20 and players A, B, C are available, in that order of preference. But you have credible intel that teams picking at 21-24 are interested in B only.
One (conservative) strategy would be take A at 20 and worry at 25 when you get there. That will guarantee you land your first choice, but also that you'll lose your second.
Another (speculative) strategy would be take B at 20 and bet that A will still be there at 25. That means you maximize your chances of getting both A and B, at the expense of a small chance of having to go with C at 25 if you got your facts wrong.
If the gap between A and B is smaller than the gap between B and C, it'd be stupid to go with the conservative strategy without considering the alternative. And though the Spurs may be conservative, they sure aren't stupid. So I have to believe they take stuff like this into consideration before making a decision.
Actually... weren't you the one arguing they took Primo at 12 because they had info he'd be taken if they traded back?