The internet that people miss was deeply interrelated with the real world. Blogs were read widely enough, even by an elderly non-techie audience, that in real-life meetings with people in a shared community (music-band fandom, a religious denomination, a literary scene) you could feel common ground with other people by following the same web content that was a locus for the community. You might even meet new IRL friends through forums, IRC channels, or blog comment sections.
That is all gone now. Followers of the indie web now are a tiny demographic, and indie-web fans are unlikely to know people in real life who read anything they read. Everyone else is on the big social-media platforms (and, for some communities, dedicated phone apps, like Komoot and PolarSteps for travel).
I think that's why people feel so nostalgic. The world and society has changed even if the HTTP protocol and webhosting at your own domain are still there.