Currently Chrome supports the URI fragment https://en.wikipedia.org/wiki/URI_fragment#:~:text=foo which will highlight the first occurrence of the word "foo".
An improved version of this would be to automatically open the "search text within page" bar when the URI contains a fragment #:~:text=<search fragment> and put the <search fragment> into the "search text", automatically finding the first occurrence of that <search fragment>
Also, add a button the the end of the "search text within page" bar for creating the fragment URI. The button could be labelled "Permalink". On hovering over the "Permalink" button, the text "Create a permanent link to this search result on this page" would appear (or similar). On clicking the "Permalink" button, the text https://en.wikipedia.org/wiki/URI_fragment#:~:text=foo would be copied to the clipboard, and a brief (2-3 seconds) message would appear "URL with search copied to clipboard" or similar.
Possibly also include a read-only box with the URI in it. The URI changes if you change the search term.
Possibly also include a way to get to the nth occurrence of the search term. e.g. #:~:text=foo#:~:count=4