Your first link appears to require the installation of an app on the phone (after installing a dev certificate). The app, running arbitrary native code, performs the actual jailbreak (most likely using a chain of kernel exploits).
The devs seem to have gotten around the App Store by using an approach like TestFlight, which allows apps to be deployed for development and wider testing purposes without going through the App Store.
I know one of the original iPhone and iPod Touch jailbreaks worked that way. I just went to a website and it automatically rooted and installed the homebrew app for me.
None of these exploits have anything to do with accessing TCP services using a browser. The only exploit there having anything to do with Safari, CVE-2016-4657, is a WebKit memory corruption (per your [1]: "The stage1 employs a previously undocumented memory corruption vulnerability in WebKit to execute this code within the context of the Safari browser (CVE-2016-4657).").
I think one of the first sites to do it was this one: https://jailbreakme.qoid.us/