I initially had a a very inadapted neon blue cable going from my circuit to the Minitel. After finding an old phone in the trash, I salvaged its spiral cable, and now it looks the way it should!

Much better!

I've finally soldered the full circuit, with the LM7805, and... it does run quite hot. Quick back-of-the-enveloppe math show that it should be fine, but... I probably need to salvage a heatsink. Or maybe an aluminum case?