For example, take this NYT article https://www.nytimes.com/2022/08/31/health/life-expectancy-co...
Google cache:
https://webcache.googleusercontent.com/search?q=cache:https:... -- 404
OP's service
https://cfworker-beatthatwall.jayass.workers.dev/?url=https:... -- works.
However, one can adopt your bookmarklet to use OP's service when needed instead of installing extension/userscript that seem to match all the sites.
> <meta data-rh="true" name="robots" content="noarchive, max-image-preview:large"/>
So, OP must be using some other means to retrieve the page.
In the case of NY Times, they're likely just grabbing the non-archived version and performing an operation similar to 12ft ladder.
Google cache fetching seems like it might be an effective strategy for a site like Washington Post that have extremely effective paywall enforcement (till your turn off JS), but also allow Google cache.
If so, you can also do this via a simple bookmarklet:
If you don't know what bookmarklets are: Edit any old bookmark and put the above line into the url field. Next time you click it, it will bring you to the Google cache of the current page you are on.