Al-Azif's provides a web server for the exploit (called "ps4-exploit-host"), that I used with a cheap USB wifi adapter.
If you want to run a different exploit, you'd need to set up a web server yourself. You can use a Raspberry Pi, a "ESP8266" Arduino board with a wifi chip, a programmable router, a wifi enabled SD memory card, or a cheap USB wifi adapter. Anything that you can link your PS4 to your PC. An Ethernet cable would probably work, too. Then you'd have to connect to 192.168..... as DNS. It's a bit tricky, but I'm sure there are guides out there. You just have to do a bit of research. It's not as easy as "put the whole folder on usb stick and re run the internet connection", lol.
And I don't understand what's so wrong with Al-Azif's exploit. All you have to do when you power on the console is to open the web browser. Done.