Because you can't. Not even an Extension is able to. Browsers don't want you to bypass their content enforcement. I wish we had at least one hacker friendly browser.
Extensions can't, correct but I wanted to bring up a special case regarding this
Isolated web apps a chrome feature for developing apps that run in chromium based on HTML (but tbh only really used in Chromebooks) do support raw TCP sockets so if this was ported to an IWA you could have Firefox on a Chromebook without an external server needed.
Extensions can inject headers, such as Access-Control-Allow-Origin: *, to unblock cross-origin requests. In the Manifest V3 context, however, that might require patching window.fetch and window.XMLHttpRequest.
This is a cors bypass but part of this demo is that it's full Firefox including TLS support. Using this still means intercepting all requests in an inspectable medium and does defeat part of the point
No one said any site should. Letting a site you control do it is a perfectly valid user choice. Otherwise people are stuck going through third-party proxies which is far worse.
Isolated web apps a chrome feature for developing apps that run in chromium based on HTML (but tbh only really used in Chromebooks) do support raw TCP sockets so if this was ported to an IWA you could have Firefox on a Chromebook without an external server needed.