Aside from the massive tether and engine that would need to pull it, I'd imagine it would rip the submersible to shreds at those depths
https://en.wikipedia.org/wiki/Jason_(ROV)
And the Titanic was discovered by camera sled being towed by a ship.
https://en.wikipedia.org/wiki/Argo_(ROV)
So your search might have been a bit too quick. It certainly is possible possible to build a tethered crewed submersible, though this vehicle wasn't.
Credit to js2 from this comment: https://news.ycombinator.com/item?id=36396161
No good for pulling it to the surface, of course, but reliable communication alone would be a great benefit.
[1] https://en.wikipedia.org/wiki/Nereus_(underwater_vehicle)
Ex.
https://www.researchgate.net/figure/Schematic-image-of-a-wor...
https://f-e-t.com/subsea/vehicles/tether-management-systems/
There's also this one which looks like it can get to the depths of the Titanic:
https://nautiluslive.org/tech/rov-argus
So it was certainly possible to do this with a tether, but it would have been much more expensive.
This sub has 4 Innerspace 1002 thrusters that make for a max total of 342*4=1,368lb of thrust... I think a useful point of reference in deciding how much pull force would be needed for an emergency recovery tether.
I think it's reasonable that a 4mm dyneema line with a breaking strength of 4,000lbs would be sufficient for emergency surfacing, and safely provide much more thrust that it's own motors. The whole sub weighs 23k lbs and is approximately neutrally buoyant.
Total weight for 3 miles of 4mm dyneema is only about 120lbs, and for simple emergency use, in the spirit of OceanGates now infamous "scrappy thriftiness" could be pulled with the type of simple/cheap hydraulic winches used for pulling crab pots, anchors, or fishing nets. 3 miles of this line would have cost about $15,000.
At 4000m under the ocean, you'd need like 10000m of cabling at a minimum I'd assume? And then you'd need the tether + winch to be capable of supporting and lifting the sub at that depth.
I'd be amazed if any country let alone company has that capability outside of maybe the US military.
They are not tethered?
Also, subs operate at close to neutral buoyancy. A tether does not need to support the weight of the vehicle, it only needs to resist the force of drag that the water exerts on the vehicle moving at whatever the max tow / lift speed is.
The vehicle that
This would cost about $3/meter.