So... Snikket? Set up a server, disable federation, and go.

What's wrong with Signal?

Not sure I understand exactly what you are looking for, but: I have a friend who hosts a Matrix homeserver with federation disabled (such as with the federation_domain_whitelist Synapse config) for children to use. It provides a reasonably modern experience, privacy, and control over things.

Note that if you want them to have calls, setting up MatrixRTC is important for having the best experience.

Phones

Rationale

Jonathan David Haidt clearly cares about the well-being of children. I can navigate to https://www.anxiousgeneration.com/resources/basic-phones-guide-for-parents from the Wikipedia article about him (through several intermediate web pages). All the resources about telecommunication that I've managed to find from a source that is directly related to Jonathan Haidt seem to assume that the primary telecommunication methods that should be used by someone who isn't yet at least as old as a typical high school student involve the Public Switched Telephone Network ("They’re a stopgap between the age when texting and calling becomes socially useful (usually in middle school, by age 12 or 13) and the age when they’re ready for a smartphone and possibly social media.").

I don't like using SMS, and I assume that all the information I transmit using the Public Switched Telephone Network is being collected by all the world's governments, but everyone who is not yet 18 years old and wouldn't be allowed to vote in an upcoming election for the next 2-7 years probably shouldn't be using telecommunication technology to do anything that would be considered interesting to a government. I disagree with the sentiment expressed with "if you have nothing to hide, you have nothing to fear", but if someone has political power they will surely use it to gain the power to influence the society they are a member of, and an obvious way to influence one's society is to gain suffrage. Therefore, someone who probably won't be allowed to vote in any election for many years clearly doesn't have influence over any government, so they won't be able to ensure that they receive fair treatment from any government, so if they eventually will be able to vote just due to waiting for a few years they should probably avoid using telecommunication technology in a way that is likely to attract attention from any government until they are able to vote in an election that will happen soon (they might be better off communicating anything "interesting" by communicating with someone in person), and therefore they probably don't need to use any telecommunication technology that doesn't involve the Public Switched Telephone Network.

Recommendations

Because this is the "Free and Open Source Software" community, I'll share some information about some "free" and/or "open source" software.

• "GrapheneOS is an open source project with an open development process." and I can navigate from https://grapheneos.org/source to https://github.com/GrapheneOS/Messaging
• I can navigate from https://f-droid.org/en/packages/org.fossify.messages/ to https://www.fossify.org/ to https://github.com/FossifyOrg/Messages

Other devices

There are some devices that can be used with a "messenger" that aren't phones, like a desktop computer or a laptop. There are also many devices that would be difficult to use to interact with the Public Switched Telephone Network.

You mentioned "Snikket or Prosody" and I see that https://xmpp.org/software/prosody-im/ and https://snikket.org/faq/#q-why-does-snikket-use-xmpp (which https://f-droid.org/packages/org.snikket.android/ can be used to navigate to) both reference "XMPP". I would describe XMPP as a "real-time communication protocol". Because of this, information around https://www.privacyguides.org/en/real-time-communication/ or https://www.privacyguides.org/en/social-networks/ might provide useful replacements for XMPP and might provide useful software that can be used with a device that is not a phone. However, much of the information provided by those pages is about software that either always allows "discovery" of people or groups that aren't subject to the review of one's parent (this surely includes Signal and SimpleX Chat and possibly Mastodon) or software that doesn't come with a clear way to self-host the underlying communication network (this seems to include Signal and SimpleX Chat and Briar). It's likely that you would more strongly want to prevent "discovery" of people that you haven't vetted personally than that you'd want to self-host a communication network, so using something other than Signal or SimpleX Chat (and self-hosting the communication network for the software your kid is using if that's possible, so that "discovery" of new people is less likely) would probably be the best way to allow a kid to communicate with someone using a device that isn't a phone.