There's a lot of anachronism in your question. The notion of a "story" that might "cause psychological damage to developing minds" would have made no sense at all to folks who'd see many of their children die in their earliest years. Death and violence were all around in early modern times; people took their children to public hangings, and church sermons were filled with quite detailed descriptions of the tortures of the damned.

The case of Struwwelpeter that you reference in your question is a bit different; it's not a "classic" fairy tale, dating to the 19th century (many fairy tales and nursery rhymes are much, much older). The author, Heinrich Hoffman, was a physician who wrote this in 1845 for his three year old son. To his enlightened eye, this was an agreeable way of teaching a child manners, and was enormously popular -- a best selling book of its day and for many years after.

It continues to echo in popular culture (think of "Edward Scissorhands") and just as a 22nd century critic would be reading Tim Burton wrong ("what terrible people they were to watch a film about a horribly altered boy"), reading Struwwelpeter today with no sense of the humor of 1845 will lead you astray.

Hoffman wasn't monstrous or cruel, and he had a particular interest in behavioral problems, so this book wasn't written to be, nor perceived at the time to be callous.

Serious critics divide over Struwwelpeter today

The divergent reactions to Hoffmann's work among adult readers and literary critics differ primarily in their assessment of its appropriateness for young readers - which is moreover determined by adult perceptions of how children will themselves react to the book. Adult critics of the book who argue that the graphic images and horrific texts will frighten young readers will thus find Hoffmann's book to be fully unsuitable for children, while those who claim that children will be amused by the comical depictions of over-the-top stories will endorse it. To be sure, adults have played a central role in the success of Der Struwwelpeter, as any work of children's literature must appeal not only to children, but also to their parents, in order to enjoy popularity on the market. Likewise, it is adult critics of the book who have been most influential in establishing its notoriety. Although adults are thus responsible for the split reception of Hoffmann's book, the central point of the debate concerning Hoffmann's work nevertheless remains its appropriateness for children

There is a _lot_ written about this book, what is going on in it, and how it is understood by the adults who buy it and the children who read (or were read) it. So if it or other (older) fairy tales seem inappropriate to "developing minds" to you, that would be your cue that the way _you_ understand how minds develop, and the influence on them of media isn't universal.

Sources:

The Influence of Struwwelpeter

A Classic Is Born: The "Childhood" of "Struwwelpeter"

Aesthetic Tension: The Text-Image Relationship in Heinrich Hoffmann's "Struwwelpeter"

Validating Defiance: From Heinrich Hoffmann to Mark Twain, Rudyard Kipling, and Maurice Sendak

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As u/amp1212 largely covered the psychological aspect of the question and the context of the specific tale you cited, I'll focus on the storytelling aspect:

Did parents underestimate how impressionable children are, and how such material could cause psychological damage to developing minds?

Largely, no. As I and others (particularly u/itsallfolklore) have detailed in previous threads on the subject, fairy tales and classic fables were not originally specifically directed for the consumption of children and merely evolved to serve that function, so the ability or non-ability of the tales to cause trauma largely did not factor into the creation and proliferation of said tales. Children were often included in the audience, but they were not the primary audience. As such, the tales had lots of violence, lots of sex, lots of bawdy jokes and references, and lots of frank discussion about issues that impacted the storytellers’ worlds. Fairy tales were chiefly for entertainment purposes, though there were often lessons embedded in them (because most tales do, after all). Writers, editors, and storytellers began to utilize fairy tales in 18th and 19th-century Europe in the moral and cultural education of children. Before that, they were chiefly for adults, whether for the teaching of lessons or entertainment purposes, which is why so many of the tales are particularly gruesome or filled with sexual exploits: they are not sanitized for children.

Fables are a bit of a different wheelhouse, because fables are intentionally moralistic stories whose entire intended purpose is to teach a lesson. Initially, most fables were also addressed to adults and covered religious, social and political themes; they became particularly used for the education of children around the Renaissance era. In the folktale/fable landscape, lessons exhibited by harsh or violent stories are just as valid and have as much teachability potential as stories with no violence at all. If the story is "the squirrel's soul was reaped by the Devil because he wasn't polite to his elders," it still imparts the same basic, fundamental message as "the little girl wasn't polite to her grandmother and got her supper taken away;" the difference is just in how much fictionalization that message undergoes for its transmission. Fictionalization gives a degree of separation through which a child is able to comprehend and work through real-life issues and lessons in a safe space, allowing them to grow up to be a psychologically well-adjusted and resilient adult. It's actually a well-documented benefit of allowing a child to read texts dealing with difficult issues as a child and adolescent.

Humans as a species are also fascinated by death and violence; children, as young humans, are just as fascinated as adults. Hearing violent tales doesn't inherently traumatize a child, especially a pre-modern child who might have been taken to public hangings on the regular or seen people dying of plague in the streets. "Developmental trauma" doesn't really factor into the fascination equation when hearing a fictional, moralistic tale meant to impart a lesson on how to behave and act. Additionally, stories like fairy tales, fables, and nursery rhymes were (and still are) one of the largest forms of cultural and political socialization that humanity has; understanding that violence, sex, and other 'adult themes' are facts of life would be seen as necessary for a child's development.