Welp, there is a way around this, but I'll respect the site owner by not posting the details.
I don't recall whether he ever put advertising of any kind on his site. If I were trying to finance the site, and didn't want to use the broader Google Ads type of stuff, I would try to sell various books or religious articles off the site, since it draws a lot of visitors.
As for the cost of running the site, it really depends on what you want to pay for in terms of performance, redundancy, etc. You can get some really cheap hosting solutions that would be more like $20 per month, but they might choke under the load. $500 per month seems like a lot for just infrastructure, since you can get a dedicated server in Microsoft Azure for about $400 ... except there's also work involved to maintain it, update it, etc. ... not to mention all the work that went into creating it in the first place. So I think what he's asking is fair.
I'm not a huge fan of the "blackmail" of taking the site down until he gets random donations. Perhaps he could make it a subscription service, or do some kind of sales/marketing on the site instead. It's also a little weak that he's asking for checks. Why not just put a PayPal link on the site?