I don't think that the cancellation of Prodigy has anything to do with systemic mysogyny, (though I can see how it can look that way).

First off, a Star Trek aimed at kids was a hard sell. Sure, it might have made sense to Paramount, seeing all the Jar Jar Binks toys that got sold, but Star Trek has always been at it's best when it's aimed at hard questions that society is dealing with right now. If it's an ethical dilemma that an 8 year old can figure out, it's not exactly an ethical dilemma. It was experiment that was tried and didn't work; unlike Lower Decks which is an experiment that tried and did work.

Secondly: TNG literally re-launched the franchise from a 3 year 1960s sci-fi serial that managed to get 5 movies (two of them good at the time), to an entire franchise, from which Deep Space Nine, Voyager, and all the rest were launched. Of COURSE Picard was more important to the franchise than Janeway.

But mostly, Voyager wasn't very good.

That's really what it comes down to. People saw Picard, rightly or wrongly, as the continuation of a character and a story that they absolutely loved - TNG. Who wouldn't have wanted another season of TNG (which we kind of got in Picard Series 3)? Now, honestly - look me in the eye and say: Do you really want another season of Voyager? Especially near the end when they were really running out of ideas?

There might have been a nostalgic draw to having Mulgrew come back as a holographic version of Janeway, but that's about it - in all honestly, the inclusion of Janeway turned me off from wanting to check Prodigy out, because I did not like Voyager.

There's also one last counter to the "This is systemic misogyny" argument and that is - Star Trek doesn't seem like a franchise to be unaware of systemic misogyny and, if anything, works to combat it. Yes, there were a lot of problems in the Brannon/Braga era (I got turned off of Enterprise with the obvious fanservice in the first episode, and what's up with Troi's first/second season uniform?), but by and large, if there ever was a franchise that took a hard look at prejudices and systemic problems, that'd be Star Trek. The joke, of course, is "When did Star Trek get so woke?"- the answer is 1966!

So - I get how taken in a vacuum, this can seem like systemic misogyny. And maybe it even is, I just don't think the preponderance of evidence supports the theory, and I don't think it fits Occam's razor.