That's for the lawyers to sort out, they have a lot of flexibility as a US nonprofit. The case law isn't that clear cut for this.

Internet Archive was almost destroyed by a copyright lawsuit from book publishers within the last few years. I expect they aren't excited to take on any additional risk of similar lawsuits right now.

They need access to sites to archive them. It's already hard to do it as it is, imagine if they start selling access to content. They'd be shooting themselves on the foot, independently of what the law says.

major [citation needed] on that. There are very limited exceptions to the massive power of copyright -- and they're mainly granted to libraries in the form of narrow waivers. And just the cost of fighting the most powerful copyright holders can bankrupt you -- especially if you're a relatively modestly-funded nonprofit.