Any caching proxy (including squid) will serve - that is their whole point. You may need to tweak the configuration to ignore the website-specified expiry and cache headers.
Squid (or any other popular caching proxy I'm aware of) doesn't cache verbs other than GET, so a lot of websites can't be cached this way; notably, GraphQL APIs usually use POST for all requests, even just queries.
In principle this is true but there are some caveats regarding the usability: It does not present the cache in a friendly way. There is no Index or something like a nice starting page with your top browsed sites - you get the idea.
I guess this is totally doable with squid or any other caching proxy, but I don't know any that does this
