There's examples of how this works in a healthy way. Martin Manley is one scenario that comes to mind, where he overtly opted-in to having an archive stored about him upon his death: https://martin-manley.eprci.com/
People who freely publish information, to the worldwide public, on the 'World Wide Web' should reasonably expect all sorts of entities to collect, save, analyze, & repurpose that info, unless they take specific steps to discourage such access & use.
The Archive's crawlers identify themselves, and collect things that are publicly linked, or specifically nominated-for-collection by library patrons or partners. Except in some focused specialized collection projects, they don't "log in" as any user, only visiting & collecting what's published freely to any anonymous person/organization/process.
For material needing more privacy, websites always have the option to block any and all unwanted visitors/crawlers with a wide variety of standard techniques, like requiring logins or simple challenges that automated crawlers won't pass.
And, as your linked articles report, the process for a later exclusion by request is pretty quick and simple. (The 2nd post concludes: "So, hats off to the Internet Archive for making the process smooth and relatively painless.") And, such exclusion does not require any sort of "DMCA request".
Thankfully DMCA should make the removal process easier now, especially in situations where control over the domain has been lost or being hosted by a third party. Although last I saw there were still artificial barriers, such as needing to list every single individual page needing to be taken down. But this is after the fact, after you discovered your reasonable expectations and privacy have been violated. And then you have to track down the other copies that IA illegally distributed your now-private and copyrighted information to, such as a few libraries around the world with similar projects.
