Earlier quoted context omitted.
The result would be that publishers charge for publication, similar to current open access options. It could be that greater usage of open access would bring down the cost, but I'm not sure. Definitely the number of grants would go down as the cost of publishing starts being factored in, although this may be partly or fully outweighed by the reduction in universities' costs.
arXiv charges neither for reading nor for publication. And for-profit publishers have to be profitable to exist, which necessarily incurs costs on authors and readers.
Regarding profit: sure, you could set up a nonprofit to do the same job. You'll end up reducing costs by 20-40%, even assuming you don't have higher costs because of lower scale/lack of experience/etc.
For other essentials like food, I've seen stories about nonprofits that get large groups together and buy in bulk. Co-ops are basically the same idea. Using a non profit to reduce costs has been done.