You wouldn’t “already” have it.

You’ve missed the point. Simply having something on your harddrive is already something the law does care about. It simply depends on the something.

Well I’ve read some copyright laws

So have I. Because I had access to an exception under it in my prior job. Seems like we’re still on the same page here. Not sure why you’d feel the need to call out someone else’s knowledge on a topic that you have no idea about.

However, Wayback Machine making read-only copies of websites has for now never been disputed successfully.

Except it has. That’s why administrators can exclude domains from it. DMCA notices also can yield complete removals.