Much of the old web is still there. A lot of old sites have gone dark, but there are still some that remain, and some have persisted for surprisingly long.
Usenet and IRC still exist! As public and distributed services, like the Fediverse and the World Wide Web itself, one node can go down but others remain. Things that don't remain? AOL IM; Yahoo Messenger; MSN Messenger; Google Talk (in its original form).
When everyone chased after social media, many people declared the old web dead. They were wrong. When mobile platforms hit it big, a lot of people thought the days of the desktop PC were gone. They were wrong too. The demise of Google Reader was an attempt to kill off RSS, but a lot of sites still have feeds. And a lot of blogs still exist, even if it's getting harder to find them due to Google Search's ongoing decay.
Corporations have big PR budgets, and a lot of tech reporters are uncritical about what they hype. Witness the attempts to get cryptocurrency, NFTs, and now LLMs, to take. But we do not have to buy what they're selling.
We don't need a new internet. The old one survives, for now at least. But we have to remember it exists, and make it easier to find.