I would also recommend checking out St. Joe River bows: https://stjoeriverbows.com/
They have a youth program for growing kids where you can get credit for trading in a working older bow when purchasing a newer one.
I know Stalker Stick bows also make some nice takedown bows for kids, can upgrade limbs as needed.
My daughters have a Bear Archery 'Bullseye X' model that is not bad for an affordable option; it reminds me of the Samick Sage that I started with about 10 years ago except it's scaled down for kids. It comes in a few poundage options and is a takedown, but I'm not sure if they sell the limbs separately so that you can upgrade.
You can get some affordable Genesis aluminum arrows from Amazon or various other places as starter arrows.