There could be various reasons:
* strong Same-Origin-Policy (serverside)
* crossorigin/integrity attributes (serverside)
* ServiceWorkers (serverside)
* missing frameworks (LocalCDN)

You can test a website online or manually:
* https://www.localcdn.org/test/check

You can always report the website as an issue at codeberg or by e-mail. Then I could see what the problem is and improve the extension:
* https://codeberg.org/nobody/LocalCDN/issues
* https://www.localcdn.org/contact