Depends on the TV. For gaming, it would be essential that it has some form of gaming/low latency mode.
Also, why would you pay extra for a "good" main board? That's literally the one thing where you can go cheap without a problem if you're not investing in the high-end segments of the other components.
As a sidenote: have you looked at something like a SteamDeck for your kid? It's a full fledged PC that you kid can hook up to the TV and if you want to watch something on it the kiddo can still use it with the build in display. the base model is also dirt cheap for what you get.