I dislike tracking as much as anyone, but can you really fault them for using Google analytics? It's not like PBS choose to use them "to profit off the viewers".
This is the problem, its just convenient and free. People are not thinking about the costs to the end consumer - which in this case is our children. I don't really fault PBS, but I do fault the advertising companies for our general lack of privacy and control over our own data. I also fault the politicians for turning a blind eye to it. I hope Apple's new strategy to go after privacy pays off in a big way and more companies fall in line with it. Or at very least more people begin to think about the trade-offs before blindly injecting google analytics into everything. Self-hosted analytics that does not feed into an advertising giant is not unobtainable goal.
Depending on the apps, you could make some age assumptions of the kid too. Even more, perhaps extrapolate the number of children in the household based on the age-ranges of the apps and the behavior of how they are using the apps. A 3 year old and and a 5 year old are going to have very different usage patterns. The number of children and their age ranges could be a very important metric when considering targeted advertising to the parents. The fact is, this information is used against the consumer to manipulate them into doing things they wouldn't otherwise do - click an ad and perhaps make a purchase. Any bit of information to further that goal will be used, even better if its information about the children since that's a major purchasing factor for parents.
In fact, so many apps embed these analytics frameworks, that it's possible to make an almost complete picture of what a device was used for every day, including location information, etc on the back-end.
GA today is much more than just pageview analytics. With something like Google Tag Manager, you can have non-technical staff add in event tracking for stuff like form completion or file downloads, without needing to get developers involved. Reporting for stakeholders is also robust because of its integration with Google Data Studio, Google Sheets and other related tools.
Ultimately though, GA wins because these organizations outsource web development and digital marketing work to agencies or contractors, and they are the ones that make the tooling decisions. GA is a known quantity, so even if their contract isn't renewed, a new agency will be able to take over the account fairly frictionlessly.
And there are more contractors that are GA-focused than there are for Matomo, and that's important when you're tasked with shopping around for agencies.
For the record, all of my projects use self hosted analytics.
But you have to understand, the marketing team wants analytics. You can request to allocate engineers to setup a server with motamo, which will require replication, setup, maintaince, etc.
Your project manager is going to say "Wait this is going to take X man hours, require maintenance, etc? Why don't we just use GA like everyone does, like we always have, as it's free, has builtin redundancy, requires no maintaince, and can be added in 5 minutes?"
They aren't going to even think about privacy. Most consumers don't even care (although they should).
To answer your question though: Instant setup, ease of use, zero-maintance, zero-cost, and it's an industry standard. Literally the only downside is less-flexibility (mostly only applicable to programmers/power users) and privacy issues, which again, not very many consumers or companies care about this form of privacy (but again they should be).