The three main methods are: acoustic using SONAR to receive data, tether or umbilical cord, and buoy based where the DSV releases a buoy that ascends to the top.
Relays don't really work because you'd need a LOT of them and they'd all have to keep themselves positioned within like 30 meters of each other which is very hard with ocean currents. That's with very low bandwidth VLF radio.
wouldn't it basically need to be the same as what they use for intercontinental cables ? tons of shielding and repeaters. maybe there'd be further risk bc if it breaks at the ship it could fall on or weigh down the submersible
Not at all. Shielding against what? It's light. You only need enough of a protective coat to make sure it's stiff enough to not twist and stretch too much and to make it neutrally buoyant. No repeaters needed, base range is about 40-60 km. ROVs going to greater depths use them all the time without major issues.
The real reason is that they were stingy as fuck and that it's mildly impractical which outweighed their complete disregard for safety.
Those also carry electrical power to supply the inline repeaters on those cables. If the cable does not have repeaters, it wouldn't need this, and the shielding could be greatly reduced.