That said, when the Internet Archive is directly competing with both the Switch Store and ThePirateBay, then lawsuits from Nintendo are basically inevitable.
I agree with your broad point, because archival is important, despite what these big corporations think, but I think that the way that IA is doing stuff right now carries a very real risk of getting sued into oblivion.
I get that your question is asking more the "how do we protect ourselves from lost media being (intentionally) created if we're not allowed to break copyright?" and I don't think the uploading to IA itself of copyrighted media is even what the media companies have an issue with. If I rip a blu-ray and store it into a private data store (e.g. a non-public Google Drive or my personal NAS or something), I doubt that the entertainment and publishing companies would have much issue. I think the issue comes from the fact that the way that IA currently operates basically has no real difference between ThePirateBay, minus the ads. Any number of people can download any number of files at a time from IA, completely unrestricted, usually without DRM, and a lot of it is media that is still actively being sold.
It would be different if IA operated more-or-less like Abandonia, where they only allow public downloads of media that has no legal way to purchase it, because at that point it would be hard to argue that they're causing any damages. If they did that, I don't think anyone would bother suing them.
I'm hardly a moral paragon; I've torrented my fair share of movies and music in my day. I've also defended piracy as a way to fight against the huge number of streaming services [1]. It's not even that I necessarily have a "problem" with how IA operates, it's that I think that if they don't look like they're making a reasonable effort to fight pirated content, they're going to get sued over and over again. Whether or not they're in the moral "right" isn't really what's at issue.
[1] https://news.ycombinator.com/item?id=39692375
ETA:
I am aware that IA did do some amount of Controlled Digital Lending, but that's not what I am taking issue with. A lot of games/movies are just publicly available, unrestricted, for anyone to download.