You seem to be a bit confused about what Tor and the Tor Browser are. Tor Browser is just a web browser with extra anonymity and tracking avoidance features (in the browser itself, and because it sends traffic through the Tor network). You visit sites you like and avoid sites you don't like, exactly like any other browser. It's not guaranteed to be anonymous: any sites you visit can record and track much of your behaviour there (just like with normal browsers), but third parties can't as easily snoop. Websites with .onion addresses are more resistant to tracking, as your traffic doesn't need to exit the Tor network to get to the destination, but Tor Browser can access most normal sites too.

I suggest you just try it.

It depends how much your friends and family care about anonymity online, because individual people's browsing behaviour can be unique and trackable - a browser cannot magically hide that. To really be confident of hiding successfully you need to do a lot more, and few people really need to or can be bothered.

E.g. I sometimes use the Tor browser at work to check whether local pages are accessible from outside the local network. There I don't care about anonymity, I just want a connection appearing to come from outside.

If someone posts CP as spam on a random onion site, it's reasonably going to get removed there too because it's disgusting and nobody wants to see that.

Moreover, knowing the passport numbers of the moderators is not a prerequisite to a site being well-moderated. It's possible to build and care enough to maintain a reputation under a pseudonym.

Meanwhile you have no actual guarantee that people won't post whatever they want on the open web. Anyone can host anything they want from any computer on the internet, including a compromised one when they don't want anyone to know who put it there. In other words, if you're willing to break the law then you can already be anonymous, so the thing Tor is enabling is for people who don't want to break the law to have anonymous communications.