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.