And LMA is 6-11 and protected the rim and rebounded better. Sounds like a center by your standard, unless you're suggesting that LMA needs someone who can't shoot next to him.
You make it sound like the great rim-protecting centers are flooding the market right now. They aren't any more available than the PFs you mentioned. The best defensive center possibly on the FA market is Amir Johnson, who himself is more of a shooter than a "traditional center". If you're talking about trades, Jerami Grant is probably available for cheap, and he's a shot-blocking combo-forward who is getting a decent three going. Either of those guys is better than getting plodders just for the of it, or taking on Henson's deal.
The Hawks don't think they're playing Horford out of position; that's the point. They aren't concerned with getting a different five. That's why they signed the guys they did. The Spurs just wanted to get the best bigs next to Duncan as they could, which is why they were keen to change so often. In back-to-back years they used their MLEs on bigs, one being a PF and the other being a center. Both guys were expected to start next to Tim. That's as clear of a precedent as any that the Spurs will focus on getting talent over perceived fit.
They were searching for a back-up center who could defend and slide into big-ball lineups. Pretty much the inverse of the minutes you were suggesting. What they wanted was a combo-forward in a good way, a guy who could space the floor on offense and defend fours on defense. The closest they got to that was Battier.
The Spurs have to maximize the spacing around Kawhi and LMA. Green and LMA's mid-range shots are just not enough.
The issue with the concept of spacing that I think most people overlook is that spacers aren't supposed to be big scorers. When Green shoots 2-5, 3-6, 1-2, 3-5, he's doing what he's supposed to do. You want guys who teams have to guard or else they give up efficient looks. That's the reason why guys like Bosh and Love are marginalized when playing next to Lebron. You can't be a star stretch-big. LMA being able to shoot a three is fine, but making that anything other than a rare part of his shot-selection will make him a less effective scorer. You don't want him spacing the floor for a worse big, and especially not for an Oberto type.