Most modern printers aren’t what you describe though, they’re either a piezo that flexes with electricity to create pressure on the ink chamber and release a precise droplet of ink or they are a thermal design where a resistor heats inside the ink chamber to create pressure that forces ink out of the nozzle and subsequently draws more ink into the chamber as it cools. Heat is used here to eject the ink but heat is not used to set the ink in either process, that is done with evaporation and absorption (which is why printing a full page image can smear).

It’s not some big secret as you’ve said, the patents are openly available, but as you’ve said they’re off limits even for noncommercial use because America is stupid. It’s true that they’re not mystical and impossible to recreate but they’re definitely harder to replicate than a heat sink with a tube cut in it, a heat break, a cartridge heater, and a metal nozzle with a (typically) 0.4mm hole

The print head in most inkjet printers (at least non commercial ones) has no moving parts (unless you count the piezo flexing). Dot matrix used needles but why recreate that unless you specifically want that for the vibes or something?