While it's true they only need a split to regain homecourt, that isn't the point. If they loose game 3, the Nuggets will start to think of themselves as the better team and the Spurs will be at a distinct psychological disadvantage.
This is playoff dynamics. The better team usually wins over a 7 game series, and they do so, by winning the psych game as much as on the court. These series only go to 7 if the teams are evenly matched and then it comes down to luck. Which is why homecourt is so big in these series. Typically in those series, both teams win on their homecourt.
The Spurs need to take control over the series and close the Nuggets out, not just to beat them, but to get through the series so they can win the next one.
If the series goes to 7, the Spurs will get taken out in one of the next two rounds depending on matchups. I say that because when a team struggles to get through to the next round and their opponent has a cakewalk, they don't have time to recover thanks to fatigue.
IF the Nuggets win game 3, they will be more than a little encouraged. They will try to close out the Spurs by doing the same thing in game 4 and they will go back to SA down 3-1. That's a formula for disaster for the Spurs, one more slip up and they're done. You don't let your opponents hang around if you can help it.
The Spurs need to establish they're the better team and make the Nuggets give up. IF they can't do that, they'll be in for the type of series that reduces their chance of doing anything significant this year.
This is also true for Dallas. They cannot afford to let the Warriors hang around, or get ideas about winning. Teams that don't close out their opponents quickly in the first are usually at a disadvantage in the next.
The Spurs have the toughest road, since if they manage to get by Denver, they will have to face the Suns and Dallas in the next rounds. They need to be at least as fresh and as rested as the teams they face if they want to win.
The last thing they need is to struggle putting out the Nuggets. That's why they need to win game 3.