EFI is ok, but, where's the installer itself?!

I'm trying to install macOS Sonoma on my HP (ouch) ProDesk 600 G3 SFF.

After 5-6 days of playing Whack an Error™️, I got into the Recovery Mode.

I clicked on install, and.... 🥁🥁 it didn't work.

I need an internet connection (so what? 700MB is the whole os?!)

Wi-Fi: External USB Ralink (in the list of don't-buys for Hackintosh) Ethernet: No router available (what!?). I can use Ethernet Tethering on my Android phone, but I need to buy a cable, so not preferred. USB Tethering: Horndis (definitely incorrect spelling) doesn't work on Sonoma (or at least for me)

Any advice?