This information stored should obviously not be time sensitive. But if you're running a client for social networks or such, it's perfectly a good idea to cache the last 10 posts seen for example, so that when the user loads the page, he instantly has something to look at while the browser is fetching the fresh content.
Local storage was designed for this I believe: a longer-life cache for unimportant data.
One key consideration is security: do not cache things on the browser that may be private or sensitive information, unless encrypted, but even then the argument could be made to not do it at all. That said, storing your application graphics and most of the UI bits in local storage can certainly improve the user experience greatly.
Use a CDN accelerator like Cloudflare to improve load time since server side caching and client to server distance can impact performance.
Caching is solved fairly well by browsers, its usually server misconfiguration or lack of that is the problem.
I think http://www.squid-cache.org/ has options to ignore some of the browsers cache headers and return cached content anyway. But HTTPS caching requires that you let the traffic be intercepted and install custom CA certs so the proxy can be a MITM caching proxy.
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[0] https://antipaucity.com/2018/07/18/a-fairly-comprehensive-sq...