Copyright was never intended to give authors ironclad control over their works in perpetuity. It was intended to give a limited period of exclusivity, in return for the expectation that the work would always become public property. Eternal copyright upsets that balance, and shackles public culture behind bars.
The rules for copyright do apply to websites. Here is what the US Copyright Office says:
"What does copyright protect? Copyright, a form of intellectual property law, protects original works of authorship including literary, dramatic, musical, and artistic works, such as poetry, novels, movies, songs, computer software, and architecture. Copyright does not protect facts, ideas, systems, or methods of operation, although it may protect the way these things are expressed. See Circular 1, Copyright Basics, section 'What Works Are Protected.' ... When is my work protected? Your work is under copyright protection the moment it is created and fixed in a tangible form that it is perceptible either directly or with the aid of a machine or device."
Source URL = https://www.copyright.gov/help/faq/faq-general.html
More FAQs: https://www.copyright.gov/help/faq/
'Circular 1' URL = https://www.copyright.gov/circs/circ01.pdf
The Internet Archive publishes the content, they don't use a robots.txt, canonical-tag (they do set a link-header) or robots meta-tag asking search engines not to index their version. For all intents and purposes, it's just a copy of your content published on their website.
I understand their goal, I'm not opposed to it on a fundamental level, but I do believe that the choice of participation should rest with the content creator.