Since you linked to Miniminuteman, I assume you are familiar with the excellent Brad Hafford of Artifactually Speaking who dunked on corrected some of Milo's mistakes in a video, and they later collabbed. Archaeology right from the trenches! (I thought I'd mention him anyway, here's a link, too)
The URLs project plays it fast and loose and archives an assortment of random URLs. This one has an IP block warning.
Some have NSFW warnings.
Other projects aim to archive a single site as accurately as possible (possibly with a deadline when the site is shutting down), so they can't afford to have their warriors blocked or rate limited. If you are, that would be because of an issue. You can choose to archive sites you don't want to visit to avoid issues.
I was going to mention ArchiveTeam's warrior because I thought it wouldn't be listed, since computing isn't really the important thing you're donating, more your virgin IP address and internet connection... but it's third on the list!
It's been a tradition not to since Garfield.