First, I'm curious as to your monitor's refresh rate. At 1680x1050, I'm assuming that it is a fixed, 60Hz refresh rate. That's about 85% of the number of pixels in a 1920x1080 monitor.
If you run across performance reviews of the RX570, they'll typically show the performance of 1920x1080, so you can assume that at 1680x1050, your frame rates will be about 17% higher, give or take.
Both of those computers you listed have a GPU that's well beyond what's needed to get 60fps at your resolution.
For the games you're playing, are you having trouble reaching that performance level? I would think that the RX 570 8GB could probably manage it, and I suspect that the CPU might be the limiting factor. Maybe RAM as well. How much RAM does your system have?
You might try running a performance monitor, that will show you the CPU and GPU utilization in real-time, while playing the games you want. The Radeon drivers have that built in, so you can bring up an overlay. It shows CPU and GPU utilization, RAM and VRAM usage, temperature, etc.
The results will likely be different for each game. But you can check whether you're seeing the GPU, CPU, or RAM being maxed out a lot of the time, and that will tell you where your weak link is in the system.
My guess right now is that the GPU is probably up to the task, but you might be running short of RAM, or the CPU might be a limiting factor.
If the GPU is NOT the limiting factor, then getting a budget system WITHOUT an add-in GPU, and carrying over your RX 570, could be the way to go. But knowing how much RAM you have, and getting some info on what's getting maxxed out while gaming, would help.
BUT... if you have plans to upgrade your monitor anytime soon, that throws this equation out the window.