The two are not exclusive like that. I mean the issue with some of these "post up" guys is they can't shoot at all, but Tim Duncan can do both. So can Cousins. Who I'm sure many would consider a post up guy. You could fit him in a system like that though. I'd pick that one point, but do support the argument.
The very best big men can score inside and out. That's what made Dirk Nowitzki one of the most dangerous scorers in the league for many many years.
That said, it's not just about the shooting ability of the big man. If you're building your offense around post-scoring, that means 25-40 shot attemps per game are going to be post-ups, and the majority of your possessions will revolve around feeding a big man in the post.
Keeping the ball moving and finding the open man in that system is going to be a lot more difficult than, say, the system the Celtics use -- whether or not your big man can reliably hit a mid-range jumper, like Cousins or Duncan. It requires your big men to be extremely skilled so that they attract a lot of defensive attention on the block or at the high post and then make really good decisions and passes in response to that attention.
A post-up system is effectively the inverse of a spread offense. In the post-up system, the big men have the greatest "gravity" and create space for shooters on the outside. The onus is on the big men to pass out to shooters or to cutters.
Spread offense, on the other hand, creates space inside by having 3 or 4 shooters arrayed on the perimeter. Those shooters individually don't need to have as much "gravity" as the one or two big men in the post-up system, and they really one need to have one elite skill -- shooting. Then the onus is on the ball-handle(s) to penetrate and either find cutters / roll men or kick it back out to the perimeter.
I think we're seeing a lot more of the latter because it's easier to find players who can thrive in that system. After all, there are more guards, wings, and undersized big men with skill and speed in the league than there are immensely strong and skilled post-behemoths.