Tim Cornell examines this story in The Beginnings of Rome page 127-128, for broader context see the 5-8 chapters of the same book.

Different sources ascribe the building of the Cloaca Maxima to different Tarquins. Plinius says it was Tarquinius Priscus, according to Cassius Hemina it was Superbus. Cornell's conclusion that the primary source of our sources was some murky tradition (be it oral or written), that ascribed the building of the Cloaca Maxima to an unspecified Tarquin king, and we cannot determine why later historians/annalists picked one over the other.

We can verify an extremely small part of our knowledge about Roman kings in general. According to Cornell it is likely that even in the age of the kings we deal with two fundamentally different kinds of government. In the first period, from Romulus to Tarquinius Priscus there was a king who had severe checks on his power. In the second period, that is in the time of Servius Tullius and Tarquinius Superbus, the former kings have been reduced to a sacral role (rex sacrorum), but the actual political power resided with tyrants who ruled with the support of the classis, the hoplite army of ancient Rome.

Furthermore, the chronology of the monarchy is extremely murky, probably there have been more kings than seven, and the age of the kings was probably shorter than the 250 years that is assumed by the annalist tradition. At least it is without historical precedent that kings in an elective (or indeed in any kind of) monarchy rule for 30+ years on average for such a long time.

We also know that annalist tradition ascribes later institutions or building projects to kings. For example the Servian walls were built in the IVth century, and the Servian organisation of centuries as we know it is also a Republican institution.

Now that we looked at the context, let's examine your questions one by one.

Why was this job uniquely horrible that it prompted mass suicide?

Cornell generally warns his readers not to dismiss tradition on whims, but alas I will do just that. We don't know if the mass suicide happened at all, and if it occurred in relation to the building of the Cloaca Maxima.

Was it common for jobs like this, especially early in Roman history, to be conducted by the citizens instead of slaves?

The slave population of early Rome was relatively small, so probably. That said we know next to nothing how the building process of the Servian Walls were organised, what taxes were levied to pay for them etc, and this was one-three hundred years later.

Was defiling a corpse by crucifying it really such an awful fate for a Roman?

Again, we know next to nothing about the age of the monarchy and the very early republic. But I can say that in the later eras which we have more information about, Romans regarded crucifixion as a particularly gruesome fate.

Or is this entire story concocted out of thin air in order to smear the memory of the king at the time, Tarquinius Priscus, and to make him seem more tyrannical?

The thing is, Tarquinius Priscus was not seen as particularly tyrannical. Among the kings known from the annalist tradition he is the last one to be elected by due process. His successor, Servius Tullius was a tyrant in the sense that he probably acquired power by "unconstitional" means, and had few checks on his political power, nevertheless he is the most beloved of all Roman kings.

As for whether or not the misdeeds of Tarquinius Superbus were exaggerated by later historians: probably. What we do know that the revolution/coup around 510 wasn't as bloodless as tradition would have us think. According to archeological evidence Rome suffered severe damages at the beginning end of the VIth century. This can be the sign that some Romans took up arms to defend the regime. There is also the theory that when Lars Porsenna took Rome (if he did) he installed the Republic instead of the deposed kings.

Cornell's opinion is that the tradition on this case is plausible, Superbus probably was a tyrant for all intents and purposes. But cruel tyrants can still be popular, especially if their cruelty is mainly aimed at the wealthy.