First, I personally (and medievalists more generally) resist the idea of retroactively diagnosing "mental illness" in medieval people. What we define as neuropsychiatric problems today are a web of symptoms we more or less agree upon exist in different permutations in different cases. "Mental illness" as a label plays specific social functions today, from permitting insurance payments for treatment to shaping media narratives of crime to affecting how we see and treat people diagnosed with different disorders. While it is sometimes possible to point out sets of permutations of specific disorders' symptoms in medieval sources, applying a modern label doesn't help us understand the Middle Ages.

But to your underlying question, were medieval people sometimes disturbed by the violence of executions and especially being executioners, the answer is yes on both counts. Medieval/early modern society worked very hard to develop ways to limit that disturbance.

First, as executioner became a designated profession with its own rules and sort of intercity fraternity (because you couldn't really have a guild of multiple executioners in a city when one executioner tended to serve a wide region) in the late Middle Ages and early modern era, executioners were social pariahs. They were typically denied citizenship in their towns, for example, and forced to live in undesirable areas. Basically, other people were uneasy around them.

Second, over the course of the early modern era, we see a trend towards judicial and practical attempts to mitigate the pain and cruelty inherent in many (not all) executions. In the cities of the Holy Roman Empire, for example, there was a gradual collapsing of all types of executions into basically just beheading and hanging except in the worst cases. Even criminals originally sentenced to the more brutal forms could ask for--and frequently did--receive "mercy" from judges.

The trappings or the spectacle of violence would remain. A criminal would be hung and then have his body displayed on a wheel, instead of being laid out with limbs placed strategically over raised blocks (not usually an actually wheel) and then having limbs and ribs systematically and excruciatingly bludgeoned before a blow to the head finally ended it. Or a relapsed heretic would be burned at the stake--except it was really just the body burning, the executioner having secretly garroted the condemned while setting up the pyre. So there was some level of discomfort with the violence of the executions, although not with the premise itself (either that serious criminals needed to die or that violent executions played important public roles of prevention and entertainment).

Third, it became common practice to get condemned criminals as drunk as humanly possible before their executions. The primary point was to make them as easy to control on the way to the execution site as possible. Of course sometimes this meant turning a violent belligerent into an ineffectively violent belligerent. More frequently, though, the goal was to induce a more passive stupor. This made things easier for the executioner (and the watching audience) on two counts. First, the physical aspect of control, obviously. But second, the appearance of resigned stoicism played into the narrative that developed around executions as a way to justify them: Christian salvation. A "poor sinner" who accepted their punishment and sought forgiveness from God was simply going through a purgative process on the way to her or his ultimate salvation. The executioner and audience could be confident in a, finally, good outcome--they were conductor and witness to a salvific act, not just brutality.

This wasn't just narrative, either. From the early modern era, we have some surviving records from executioners and from pastors assigned to minister to condemned prisoners in the weeks leading up to their execution. These records display a deep concern to see signs of repentance and acceptance in the "poor sinners"--especially in cases like young women convicted of infanticide. Executioners and chaplains note deep relief when these women would finally repent--which suggests when there was no repentence forthcoming, they were not entirely content with carrying out their necessary role.

(And, yes, in some cases executioners also record their frustrations with "special grace" or less painful execution methods handed down by judges for particularly evil criminals.)

Should you chance to read early modern German and possess a love of Fraktur, one version of 16th-century Bamberg/Nuremberg executioner Franz Schmidt's execution record is online through Google Books! Should you not, a recent and very cool book on early modern executions is Joel Harrington's The Faithful Executioner, which uses (a slightly different--more expanded--version of) Schmidt's records as its major source.

Follow up question: how did one become an executioner? Im assuming it's not a job for which one would out right apply.