The practical viability of the cut for the rapier should be evident from the fact that it's in the period manuals. Why write it down if it didn't work? Sweetness was one of the English Masters of Defense, one would hope that he knew what he was doing (not that England was renowned for its rapier styles). Similarly Giganti details how to use the cut.
We are in complete agreement that it is definitely not a primary action of the rapier, but it is a useful situational tool. It does not make large, sweeping cuts akin to cutting swords, it makes short snappy cuts to wrists etc. (things with exposed tendons, mostly). These actions do not leave you vulnerable after the cut, if the cut misses. To be honest the main vulnerability lies in the attack into the preparation. There are ways to mitigate that threat but it is difficult.
If speaking of a cut to mildly wound, the disadvantages do diminish greatly. A cut that you describe does not require one to leave the line of attack or defense but has the notable disadvantage of also being unlikely to cause any significant wound. A small amount of sturdy clothing could easily negate such a mild attack. By contrast, making similarly small motions of the point along with a thrust has the notable advantage of being all but assured of causing a would should the blow land - less than a pound of pressure is necessary to achieve that end at the point.
But, again, from a practical perspective this does produce a problem. In order to achieve such a cut, one must be disconcertingly close to the opponent - a single fencing action from the wrist and hand. Delivering such an attack by a cut would require one be a fair degree closer than the attempting to hit the same target with a thrust and thus a greater commitment to the attack.
And, for the record, pointing to the English when it comes to the Rapier is folly given they were some of the last Europeans to give up the broadsword. The greater utility of the rapier for personal combat was eventually recognized, however. Of course, that the great masters of the art and all of the developments we remember (save for the court sword which was as much a political move as a practical one) came from other nations giving rise to broad competing schools of thought from the famous Spanish circles to the Italian and french disagreements regarding how best to hold the blade. The Italians, for example, are more likely to favor a cut given their development of a style of grip that sacrificed fine point control in favor of power where the french more clearly favored the thrust with a grip that sacrificed power in favor of fine control.
To put this in perspective, a properly held rapier utilizing a purely straight grip is actually held and maneuvered with two fingers - sufficient to direct the point but a problem if your attempt to laterally direct the weapon to some direct offensive end. The Italians, on the other hand, favored a grip that used at least three fingers in a grip that was quite solid. This had the effect of changing to the wrist for fine maneuvers (a less precise choice) but ensured any such action was backed by more strength than the french grip style could muster. The argument of superiority of style arising from this fundamental (and seemingly minor) difference raged for ages even into the modern sport until, eventually, the orthopedic grip was developed which was functionaly a compromise between the two competing styles.
Now, as to the continued existence of the cut in manuals, there are a few schools of thought. First, while the development of the rapier eventually lead to a weapon more or less wholly incapable of delivering an effective cut, that isn't to say it was entirely incapable. Blood could easily be shed even if a great many cuts would be required to cause a fatal wound. Given that duels eventually transitioned from affairs to the death into ones where first blood sufficed for most (even though such a thing stands in direct contrast to the most notable manual used to govern such affairs), the cut could remain a valuable tool. But, that said, there exists an incredibly strong argument against the cut that we find in the parry. Over time, the parry was reduced to nine fundamental manevers. Of those nine parries, you can see an inherent weakness to the cut
Prime protects one's high inside line and the position can best be described as looking at your watch
Seconde protect one's low outside line from rising attacks in this line
Tierce protects the high inside line and is notable because its only utility is to proct the wrist from the small cuts you advocate. It does not exist in modern foil for any practical purpose but it remains in epee as attacks on the wrist are valid
Quarte Protects the high inside line - also the standard position from which most fencers choose to start when commanded to be on their guard. Parries in this line can are as often circular as they are lateral
Quinte Protects the high line, notably the head, from falling attacks. This is one of the most common parries in sabre but it has virtually no use with a thrusting weapon.
Sixte - the same as quarte only for the outside line. A lateral parry between quarte and sixte (or vice versa) is the most common parry seen in sport fencing and often the only one many sport fencers actively use
Septime - also protects the low and outside line from rising attacks
Octave - protects from low inside attacks
Neuvieme - Similar to septime but protecting a higher line. Rarely used or taught given that it is broadly similar to a parry in quarte.
Notably absent are attempts to protect the head from cuts arriving laterally from the inside (prime protects outside) or any strong parry to the low outside line that would be necessary to protect against the cut. This means that defense against the cut in many lines would require an elaborate circular parry which stands directly against the economy of action that western swordsmanship began to advocate - that is, that the French advocated. This places our argument at an awkward impasse - we can see from this directly that the cut was seen as little threat based on the fact that while one constructs a full cage of steel against the thrust, there exist glaring openings against the cut. Indeed, many of the parries in this that protect against cuts appear to only do so by happenstance! Simultaneously, however, this weakness means that the cut is both unexpected and difficult to defend against.
But, as you said, we are in broad agreement I think. Yes, the weapon had an edge and yes it could be thus used to wound with a cut. Yes, manuals do indicate it's use even when the development of the weapon lead inexorably down a path that produced weapons that increasingly advocated the thrust over the cut that largely retained an edge to dissuade people from grabbing the blade. So, we simultaneously have a case for the cut and against the cut and little record of the practicality of the move. I suppose we will have to be satisfied with leaving this discussion at little more than the fact that I do not think the maneuver would often be a prudent choice in a duel. It is possible that my own position is biased based on the fact that the modern sport favors the French school of thought rather than the Italian. Were it the other way around, I would probably look more charitably upon the cut.