The issue is that all of the GPUs you have tested are consumer models (GeForce models). Pixel Streaming with Nvidia cards requires cards that support NVENC (Supporting documentation here) to encode the video to send to the browser.
As you can see here, most consumer Nvidia GPUs can only support up to 3 concurrent NVENC sessions, hence why you have found 3 to be the maximum number. If you need hardware that can support more than 3 concurrent sessions, you will need to look at the Quadro and Tesla GPU lines, both intended for professional use.