Are there areas where only some universities pay their sports scholars?
In the United States, many (perhaps even most) institutions do not give sports scholarships. 

For example, I went to MIT, which gives only need-based funding (0% sports scholarships), but has a quite high NCAA varsity participation rate because, well, sports originally exist because people enjoy playing them, especially in the company of others. That's certainly why I was a student athlete (even captain of my team). 

What institutions without sports scholarships tend to lack is not athletes but audiences.  At MIT, for example, the varsity football games were infamously said to be better attended by the band (who loved chances to play) than by anybody else, though I can't verify this because I never bothered to go myself.

By contrast, the institutions that do give sports scholarships tend to have big crowds and be the high-visibility teams that you see on TV, thus exaggerating their apparent frequency.

Given the wide mix of public and private institutions with varying philosophies and missions, I expect you'd likely find this contrast in every major US city.