. . .
Those guided cenote dives are widely believed to be safe, judging by their frequency and publicity. And the reason I think they are believed to be safe relates to the conventionally made distinction between "cavern dives" and "cave dives".
. . . The exit (or at least the direction of exit) should be visible at all times. The divers must remain close to open water. . . .
Yes and no. As someone else pointed out, the reason these guided cenote dives are not prohibited outright is that the local dive community shoehorns them into the definition of "cavern dive," which much of the worldwide community does not prohibit open water divers from doing. A generally accepted definition of cavern zone is that it extends no farther than the daylight extends, up to some distance that's close enough for a diver to fairly easily swim to open water, no restrictions, etc. As has also been pointed out, in Mexico the definition of "cavern" is stretched to its limits and beyond in order to encompass many of the cenotes where these guided "cavern" dives take place. There are at least some stretches along the cavern lines of some cenotes where daylight is nonexistent or so dim as to be essentially nonexistent, and which would be a longer swim back to open water than the distance by which the major training agencies define the cavern zone. (Since the line may be circular, and there are places to pop up into open water here and there along the route, it usually isn't immediately apparent to the diver which way leads to the nearest open water.) No, there are definitely stretches along the cavern line in some if not many cenotes where these dives take place that are beyond what the major agencies consider cavern zone.
As has also been pointed out, there are wrecks out there that similarly stretch the definition of cavern zone, and in which open water divers are routinely offered trust-me dives by guides. I suppose if those wrecks became as highly visited as the cenotes have become, and a proportional few deaths occurred on those, we would have a thread discussing similar issues.
I've gotten as far in my training as a cavern course, and I don't see how that alone makes me substantially safer in these guided cenote dives (other than the fact that since taking the cavern course I have not yet decided whether I am going to do another one; I feel more educated about the risks). A tiny bit safer, maybe, but every dive we do to gain experience, every extra minute of thought we put into the plan, gear check, etc., makes us a tiny bit safer. If a guide gives a thorough cenote dive briefing according to the local guidelines, it's sort of a (horribly) abbreviated cavern course, covering the rudiments of light use, swimming as a team, following a line, non-silting kicks, etc. (And of course everyone "passes" the "course"--no one is denied the cenote dive they paid for.) The guided cenote dives probably have become safer since the guidelines were adopted. Do the guidelines go far enough? Are too many people still dying? Where should the line be drawn in determining what is necessary to keep divers as safe on one of these cenote dives as any other cavern dive elsewhere in the world? There is no way that guided cavern (including wreck) dives will ever be banned completely, worldwide, such that all divers must be cave certified. Require cavern certification? Or something more practical, like adding a sweeper guide in the rear as mentioned several times in this thread?