Client-side storage (cookies, cache, etc.) is one of many things worth disabling during site testing. The Tor Browser’s “safest” level blocks SVG, remote fonts, JS, and other features; many textual browsers don’t support anything besides a subset of (X)HTML. Most non-mainstream search engines are similar.
Personally, I try to keep sites curlable. It should be possible to use a plain curl URL command to get all the necessary markup, complete with visible contents. If the main content of the page is text, then everything else should be a progressive enhancement.