The even the AMD R5 3600 would technically be an upgrade, offering competitive or better performance in straight gaming and being much better for streaming or video editing.
The R7 3700x would be great. 3800x only really worth it if the price difference is not too much locally.
A B450 mobo is fine. If you don't mind spending a bit more, the Asus X570 Plus has a lot of features and PCIe 4.0, without being outrageously expensive.
AMD is definitely the better bet for now, thanks to the extra threads and/or cores at a given price point, which will be an asset for streaming, and the gaming performance is very good.
For the PSU, Seasonic Focus Gold / Plus , Corsair RMx / RM2019 depending on local pricing. TXm is pretty good and is sometimes cheaper. PSU availability right now seems to be all over the place.
You can use https://pcpartpicker.com/
to help you comparison shop pricing wise, and you can post a list of parts here if you want feedback.
What model PSU did you have before out of interest?
As for the GPU, what resolution and refresh rate does your son game at? 1080p 144hz..?