Sure. « Dead » CRTs often mean dead, or ill analog boards. But on that particular one, the tube's neck has been obviously broken. No hope on that side; that machine has been used by tons of girls and boys from a public school. It has a big « 16 » mark on the side from an inventory sticker, it's completely yellow and the switch area is dirty as hell. So I would have been glad to be able to use its logic board, but otherwise I don't really care about the case and repairing the analog board…
So, a faulty logic board is sure enough the issue. Doesn't matter a lot, since I'm really too bad at the soldering iron to try anything…
By the way, thanks for the help!