It’s all about copyright. Copyright law in the US gives a monopoly on distribution of copies of things (hand‐waving because the definitions are hard, basically artistic works) to their author. Of course authors usually delegate that right to their publisher for practical and financial reasons. There are some fair use exceptions, but this basically makes it illegal for anyone else to make and distribute copies of the author’s work. Again, hand‐waving because I don't want to have to write a dissertation.

When IA shows you what a website looked like in the past, they are reproducing a copyrighted work and distributing it to you. In some cases, perhaps many, this is fair use. IA cannot really know ahead of time which viewers would be exercising their fair use rights and which would not. Instead, IA just makes everything available without trying to guess whether the access would fall under fair use or not. That means that many times, possibly most of the time, IA is technically breaking the law by illegally distributing copies of copyrighted works.

But _owning_ a copy of a copyrighted work is never prohibited by copyright. It doesn’t matter how you got the copy either.

Therefore, pretty much any time someone asks for something to be hidden or removed on copyright grounds, they go ahead and hide it. They don’t bother to delete it though, because copyright doesn’t require them to. If a copyright holder asks for it to be deleted then they are overreaching, and should know that any sane person would object. But as far as I am aware IA doesn’t actually bother to object in writing; they just hide the content and move on.

This means that researchers can visit the archive in person and request permission to see those copies. For example if you are studying the history of artistic techniques in video games using emulated software on IA, you might eventually notice that all the games from one major publisher are missing (except iirc the original Donkey Kong, because they don’t actually own the copyright on that one). You could then journey to the Archive in person to see the missing material and fill in the gaps in your history. Or you could just ignore them entirely out of spite. This is no different than viewing rare books held by any library, or viewing unexhibited artifacts held by a museum, etc