Well I've now actually finished the first season (hopefully not the whole series), and I'll say that, while I enjoyed it once it got past some of the early X-character-needs-to-learn-Y-lesson episodes, I have a bold theory as to why the show has had underwhelming results: it's not quite there as a children's show.
Put another way, as a Trek show for a family with young children to watch together I'd give Prodigy season 1 an A+. As a show for adult Trek megafans to watch on their own I'd give it a B. As a show for children to watch on their own... C+ maybe?
Basically if you were an actual child in the intended age range and unfamiliar with Star Trek then this show drops way too much difficult and abstract exposition way too fast in almost any given episode, and expects you to retain knowledge of much of it to continue with the show. When, for example, the show encounters the Borg it throws the very unusual concept of what that species is at viewers while also explaining why the crew has to go on their ship and the concept that they are currently asleep, all in one scene. Each of these concepts is expressed in both a Trekian technical vocabulary and a simpler gloss, which is good in principle, but I think it simply all goes by too fast, probably even for the average adult, if the viewer doesn't already know what the Borg are, and the need to have Trekian technical vocabulary just increases the number of words the viewer who doesn't understand that has to sift through to make sense of the situation. While "the Borg" had enough cultural impact that even most adults who've never really watched any Star Trek could probably tell you roughly what they were, that sort of diffusion of pop cultural knowledge tends to not work as consistently with children.
Another episode feels the need to explicitly explain the "Fox-Hen-Grain" puzzle (which is not actually relevant to the plot), but then explains its actual premise of the crew being separated by time dilation which expanded out from the warp core in a sinusoidal wave in a very rushed exposition dump, and centers around the need to build a "warp matrix" macguffin. While children are generally more adept than adults at putting up with things they don't completely understand in media, they are usually also less adept at recognizing what things they aren't actually expected to understand. Trek fans have developed a certain level of skill for intuiting when something being discussed is a technical thing they can or need to understand to follow the events of an episode, and when it is just techno-babble being dumped for worldbuilding sake, but I think Prodigy assumes too much of this skill in their younger viewers. While you and I know instantly that we don't actually need to understand what a "warp matrix" is, I'm guessing many children would only later on in the episode figure this out (as it is gradually made more explicitly clear that is is just a doo-hicky you don't need to understand), possibly giving-up in frustration first. I think the show really needs to rethink how it approaches techno-babble, and be more explicit about what stuff is actually important for the viewer to understand. A couple lines here or there could make it much more explicit that "you don't actually need to understand X or Y, because the actual important concept is Z".
The villain's motive centered around the rather advanced concept that his society had torn itself apart in factionalism in response to First Contact with the Federation. While the show did a great job of repeatedly drilling into the audience what First Contact was, the "socially driven societal collapse in response to foreign contact" aspect is really pretty demanding of children in terms of social studies knowledge to imagine how that would happen, and they only mention it a couple of times without showing us what that looked like. A more readily comprehensible scenario which functionally gives the same motive, and which many children would vaguely know the historical parallel for, would be that the Federation accidently gave them space smallpox. In any case, while I applaud them giving the villain a subtle and challenging motive, I think since they did not have an extended segment showing us how the societal collapse proceeded, it was probably too advanced a concept to just throw out there the way they did, for a core plot point.
Ironically, while the energy, speed, and visual presentation of the show are certainly more appealing to most kids than any other Trek, I think the average episode of TNG was probably easier for an uninitiated child to make sense of without parental guidance, because difficult concepts were introduced at a slower pace with more repetition, and no assumption that the viewer remembered the prior episode.
Even if a kid is able to follow the episodes with no trouble, homages to this, that, and the other Trek thing are a lot of the appeal of the show. Those don't work for you if this is your first Trek.
So what I think we have here is a great show for a child to watch with a Trek-loving parent who can explain things. It also would work for a child who has watched and at-least-kind-of understood a fair amount of other Trek to watch on their own, but I suspect even at that point we are getting into people who are getting a little old for some of the more kiddie-friendly elements of this show. Ultimately, I think that the child audience for this is probably too niche an audience to support a show with any real budget, and that not enough of the audience of adult Star Trek fans is going to put up with kids-show styled Trek when we are in the midst of a Trek glut. If the show was going to be successful I think the scripts needed another pass, probably with child development experts, to be tightened up to be more accessible as a "My First Star Trek".
Which is all a bummer, because I'm sure when I was 10 I would have absolutely loved watching this show with my Dad. And I don't think the issues as a children's show were by any means unsolvable, as it really comes down to just rethinking the writing (and pacing) around exposition of technical things and Trek legacy things to be less demanding of young viewers new to the franchise. The characters and core premise work, and if there's one thing early TNG is a posterchild for its that when a show's characters and core premise work there is always hope on the horizon that the show will become great. But while it may have been a triumph as a family Trek show, in its capacity as a show for a substantial audience of children to watch on their own I suspect Prodigy at least somewhat missed the mark.