That sounds like an interesting idea. I like the idea of capturing all the text content I view on the internet and building a full-text search index. Would also be fun to play with NLP and machine learning. I run Tailscale on all my devices, so it would still be easy to use it when I'm away from home. I already have a Linux server that I use for Home Assistant, so I could set it up on there.
I'm just not sure how this would work with SSL. I might have to install a self-signed SSL certificate on my devices, but it wouldn't work if a site uses certificate pinning. Is there an easy way around that? Maybe run Chromium with some CLI flags, or use a fork?