I would say so, yes. The 6c/12t is going to be a much better option vs 4c/t if you want to do anything other than *just* game (ie even multi-task, depending on the title). While the full 6c/12t may not be utilized, depending on the streaming method used etc, it's certainly going to be utiliing >4c/t.
FWIW, I wouldn't suggest buying everything from one place (unless you're getting the assembly service, where you have to).
A Ryzen5 1400 is $197 on Amazon
https://ca.pcpartpicker.com/product/72wqqs/amd-ryzen-5-1400-32ghz-quad-core-processor-yd1400bbaebox
A 2400G is $199
https://ca.pcpartpicker.com/product/FGPKHx/amd-ryzen-5-2400g-36ghz-quad-core-processor-yd2400c5fbbox
That'll be true of other components too, it'll be cheaper overall to shop around. But make sure the motherboard is "Raven Ridge Ready".
As for the GPU, no, I wouldn't drop from a 1050TI to an RX560..... at least not unless the price difference was huge.
A 1050TI is, without a doubt, the stronger GPU.
A 1050TI should be available for <$250? I can only find the cutdown 896 RX560's available, and they're >$220. Not a chance.