As I read the Chronicle's piece I don't see where schools would be required to stop giving scholarships in non-revenue sports, just that it's likely that they will.
I suspect that schools which have done really well in track or cross country, the Oregons, Stanfords, LSUs, maybe Colorado and Wisconsin, etc, will not want to de-emphasize sports where they contend for national titles and they're all in the money conferences anyway and shouldn't need to cut scholarships.
I suspect that marginal schools, ones with no great track and cross country traditions and not a ton of football money coming in will either drop some non revenue sports or do away with scholarships in those sports and use the money to try to pay their football and basketball players hoping to be competitive with the Big 5.
Then you'll have the conferences which don't give track/cross country scholarships now, Northeast Conference, Patriot League, etc. where the sport really is recreational and probably helps increase the number of applicants some schools get each year. They likely won't be affected much at all.