Government does not perform "the same role" of building up social capital at a truly grassroots level, unfortunately. Not even close. And the jury is still out wrt. non-religious social organizations, though some do appear to help quite a bit along those same lines, viz. labor unions and perhaps similar "interest-focused" mass groups.
"The jury is still out"? On what, the Rotary club?
Your underlying assumption that churches are unique as providers of social capital in modern society is... peculiar. The fact that churches are tax-exempt is a consequence of the dominant position of the Catholic church in the middle ages, and the compromises made in the subsequent struggle to remove the church bureaucracy from the government.
I expect that as religiousness across the US population declines, churches will gradually be required to behave more like other non-profits. But that will probably take many decades.
