I hate the idea of having to pay for a service in order to use something I already bought.
What do you mean by that?
Things like Xbox Live and PSN Plus. It's like I'm paying to use my own console.
That make sense, but try to think of it from a business standpoint. You would have to pay to purchase hardcopy  games without those services. They provide you with many games that you only pay a fee to use. It's a really solid business model but I understand not wanting to pay for them.
The problem is that I have to pay every month to keep access to them or I can't use my console.
Yeah I guess that's just the tradeoff. You could start investing in the hardcopy, but I think you'll end up spending more money.