As a rule, they absolutely aren't.
In this specific place, PADI allows some of them to make a supervised dive at a certain site following a specially installed line. This doesn't change the rule, which is "Open Water Diver certification does not qualify you to dive on shipwrecks, nor caves".
At some other place, there may be an option for a person with Open Water certificate to drive a semi-trailer truck, while under supervision from a trained professional, and over a course specifically crafted to minimize any possible collision. This doesn't change the rule, which is "Open Water Diver certification does not qualify you to drive semi-trailer trucks."
Where I'm confused is the difference between a direct line of ascent (even when there is an overhead directly above) versus penetrating beyond that. This is a differentiator that makes swim-throughs OK but most caverns not OK.
Edit: Turns out I can't edit a comment that old.
> certified Open Water Divers can take a cavern tour guided by professionals qualified by some of the cave diver training organizations to lead such tours. This is a highly controlled, limited tour along a line into the cavern zone, which is defined as within the natural light zone and within 130 to 200 ft (39-60m) (some minor variations here) of the surface.