“I wonder why the vessel wasn’t tethered by a cable to the support ship.”
Well, like the story says “we don’t have a 13,000 foot cable”.
I imagine they could have one made, but they were set on building an independent submersible, not a diving bell. And maybe the cable would have cost more than the vessel they actually built.
I'm not an engineer but creating a cable tether would require a gigantic winch. It would have to be able to lift the 11,000 pound weight of the submersible (less any residual bouyancy), the weight of the 2+ miles of stout steel cable, and the force needed to rotate a huge cable drum, all the while overcoming any resistance the water imposes on the submersible which doesn't look very streamlined.