The only way we own a web of our own is to develop much more of a culture of leaving smallish machines online all the time. Imagine something like Tor or BitTorrent, but everyone has a very simple way of running their own node for content hosting.
That always-on device? To get critical mass, instead of just the nerds, you'd need it to ship with devices which are always-on, like routers/gateways, smart TV's. Then you're back to being at the mercy of centralized companies who also don't love patching their security vulnerabilities.
This is very right. There are two obstacles.
(1) Security. An always-on, externally accessible device will always be a target for breaking in. You want the device to be bulletproof, and to have defense in depth, so that breaking into one service does not affect anything else. Something like Proxmox that works on low-end hardware and is as easy to administer as a mobile phone would do. We are somehow far from this yet. A very limited thing like a static site may be made both easy and bulletproof though.
(2) Connectivity providers should allow that. Most home routers don't get a static IP, or even a globally routable IPv4 at all. Or even a stable IPv6. This complicates the DNS setup, and without DNS such resources are basically invisible.
From the pure resilience POV, it seems more important to keep control of your domain, and have an automated way to deploy your site / app on whatever new host, which is regularly tested. Then use free or cheap DNS and VM hosting of convenience. It takes some technical chops, but can likely be simplified and made relatively error-proof with a concerted effort.
Both or those are solved by having a tunnel and a cache that is hosted in the cloud. Something like tailscale or cloudflare provides this pretty much out of the box, but wireguard + nginx on a cheap VPS would accomplish much the same if you are serious about avoiding the big guys.
If you already pay for a cheap VPS, why not host the whole thing there? It's the simple Web. (As has been noted in comments elsewhere.)
if only we all had a little device that was always on and and connected….
If I'm reading the implication right, you're having a pretty terrible idea. Glossing over what running a server would do to your battery, it would never work because of the routing issues you'll run into.
With IPv6 it would theoretically be possible, but currently with ipv4 and NATs everywhere, your website would almost never be reachable, even with fancy workarounds like dynDNS