https://developer.chrome.com/extensions/nativeMessaging
https://wiki.mozilla.org/WebExtensions/Native_Messaging
That bypasses the entire certificate question and lets the website know it's communicating with exactly this app and not something happening to listen on the port (and vice versa, too).
... Or, depending on what you're doing, just use a real desktop app, perhaps with an embedded browser.
It sounds like random other websites (Ebay, etc) would be able to interact with people's USB devices this way too. Maybe without people knowing?
The browser connecting to the government website accesses two servers: the original one and the second local one you install yourself on your system. The local server runs natively and therefore can access the USB device. Like all servers it should be programmed such that misuse by hackers is prevented.
The only thing missing is a rogue website abusing it.
There's no guarantee you will never connect to any rogue website that abuses this government mandated backdoor.
https://en.wikipedia.org/wiki/Kazakhstan_man-in-the-middle_a...
This website used to run Java applet. This applet was signed and it could access restricted APIs to access USB device. So website talked to applet and applet talked to USB device to sign data.
After major web browsers disabled Java applets, they implemented another approach. Now user must install a software which runs a web server on 127.0.0.1. This webserver listens on a specific port and uses web socket to communicate.
So government website now uses JavaScript to connect to 127.0.0.1:12345 using websocket. And then it uses that connection to interact with USB device.
So an ability for external website to connect to 127.0.0.1 actually is essential for this use-case.
My guess is that there are plenty of other websites which use local web server to interact with locally installed software. I know at least one another such a website: Blizzard website. It runs web server in its game launcher and website can communicate with it.
PS also they have to install custom trusted certificate because browser requires wss from https and there's no easy way to get a legitimate certificate for that kind of use.