What we really need are updated copyright laws that allow for research and preservation. However, people have been calling for updated laws for decades, but powerful copyright holders have stood in the way, so don’t hold your breath.
If you did, you could DMCA claim it. They'd still archive the stuff, but it wouldn't be publicly available until whenever it'd fall into the public domain. That part of the system seems to work fairly well, except the century it would take for it to fall into public domain.
I picked three random pieces of recent copyright content (Adobe Lightroom, the video game Hollow Knight, the Taylor Swift album "Folklore"), and I found all of them easily and trivially on the Internet Archive. In fact I found hacked versions of the entire Adobe CC suite. The IA is at least as complete a source for pirated content as most public bittorrent trackers.
Officially, of course, the Internet Archive is subject to DMCA safe harbor protections. This is user uploaded content, and so long as IA responds quickly to takedown notices, they're in the clear. But this story represents the situation de jure, not de facto. Ever since Viacom v. YouTube, it's been abundantly clear that even the biggest online platforms can't use this excuse in practice when sharing of copyright IP becomes rampant. YouTube put in the first version of their much maligned "Content ID" system the year after the Viacom lawsuit, and the parties settled out of court after YouTube lost at the circuit court level. [1]
I don't believe that the Internet Archive, as a much smaller entity, is likely to see a significantly different outcome - that is, unless the lawsuits manage to bankrupt them entirely. This would suggest they need to be much more proactive about removing or disabling access to copyright content on the platform.
[1] https://en.wikipedia.org/wiki/Viacom_International_Inc._v._Y....
A lot of what they make available arguably shouldn't be protected by copyright under a rational copyright regime but that's an argument to make to Congress rather than saying the IA can do whatever feels justified to them. (Which can cut both ways once you say the law is whatever you personally think is fair.)