While I agree that the rapier is not the light weapon most people are expecting, I also think you're over-stating the ineffectiveness of the cut. It's certainly not a primary tool of the rapier, but it is there and it can work effectively for disabling blows (unlikely to be useful for fatal cuts). As for missing with the cut - it's really not that hard to roll up into some kind of hanging guard as you retreat, or if you continue in to come to grips.
The problem with a missed cut stems from the length of the blade. In order to quickly redirect the blade to any useful end, this would require a cut along fairly specific lines lest the weapon simply end up on the ground rather than allowing a rapid follow through to the next fencing action. These lines, by and large, would either be far too grand and easily parried (directed at the head for example), or would be maneuvers that simply appear to be cuts but are, in realty, primarily used as attacks on the blade itself (the moulinet for example).
The problem with each of these is that without some preliminary action clearing the blade from a line, this leaves the attacker vulnerable to a simple parry-reposte (parry one, four and five suffice for cuts provided a slight modification of one to allow protection of the head) which should suffice to demonstrate the problem with a cut; the thrusting weapon spawned eight classical parries and only four are necessary for complete defense against the cut while all eight are necessary against the thrust). More troubling, since any cut would by necessity be a moulinet (the quickest way the weapon can be returned to line), any parry leaves the defender in perfect position to reposte as every classical parry has an attack in the same line! As such, the cut simply cannot be the primary method of attack with such a weapon though I can conceed it's use for attacking the blade as part of a complex attack risky though it may be. It has at least a single advantage in that it would be relatively unexpected precisely because of the risk and if used to attack the opponents blade (to move it out of a line), the surprise combined with the relative difficulty both fencers have at returning to any given line may prove sufficient to deliver an attack.
But, spending such time on the cut without at least acknowleding why it is inferior by explaining the utility of the thrust is probably a mistake. As such, it is relatively easy to explain why the thrust is superior for such a weapon. First, the blade can simply be presented (the arm extended) and delivered (via a lunge or other piece of footwork) far quicker than a cut. Second, the thrust originates naturally from a position of parry allowing an easier transition from defense to attack. Additionally, because there are more lines of attack for the thrust with such a weapon, it is comparatively more difficult to defend against. Finally, a thrust can relatively easily have it's line of attack changed mid attack allowing for one of the easiest complex attacks but one that has historically been the most effective (That is, you feint in some line of attack drawing a parry only to disengage and resume the attack in a different line) ways to deliver a blade.
Weapons that were truly developed around the cut lack the weaknesses described above making it not only a viable means of attack but even the most generally useful one. The sabre, for example, allows for a far more aggressive style of attack, owing in part to the utility of the cut and the ease of delivering a fatal or debilitating strike. The cut is, indeed, a wholly useful maneuver when the weapon in question was designed around the act. The Rapier, by contrast, was derived from a cutting weapon but the changes inherent undermined it's capacity to cut while simultaneously saddling it with a host of problems that would take a several centuries of development driven by the pragmatic needs of survival and utility. Seeing where the weapon ended up, as a weapon useful almost entirely for a thrust, light and not reliant on any other implement shows us handily that the problems I've outlined were seen as a problem by many.
I don't have much to say about sport fencing, because of the kinds of rules you mention. At this point there's not much of the original art in it, other than some of the techniques and a subset of the principles. Which is not to say it's not a great sport, I'm sure it is, I just have no interest in it other than as a sport.
Sport fencing has a handful of esoteric rules that make little sense from a pratical standpoint. In sabre, one cannot cross over one's feet - in other words, they may not break into an actual run. Most of the silly rules actually have their roots in praticality. Rules of right of way in foil point to the weapon's roots as a training implement; after all, if your life is on the line, it is generally prudent to defend yourself before attacking in return. Target area rules were likewise derived from practical purpose. In dueling, aiming for the face and head was looked down upon leaving the torso as the only vital spot for a thrust. Sabre, meanwhile, attempts to mimic the target area of a fellow mounted soldier (the waist up). Epee, on the other hand, discards most of those rules giving a freedom that generates an incredibly conservative fencing style. While it seems odd, rules of right of way and restrictions of motion produces, somewhat counterintuitively, a style of fencing with more freedom of action both on attack and defense.