I disagree. The 5D will give you an immense increase in quality compared to a webcam, and it'll handle ISO noise on a far superior level. Sure you might not be getting the max performance, but even after compression it's going to look far better than a webcam. If he's got them already, I say go for it. Whether you should buy one exclusively for streaming is a different question all together.
To answer OP's original question, I've had great success using multiple Razer Ripsaw capture cards capturing the HDMI output of my cameras. Delay is almost imperceptible beyond what the camera is doing itself. Quality is near that of a video recorded in-camera.
For your wireless needs, I'd recommend the Aputure Array Trans. It seems like a great budget option although admittedly I have not used it myself. https://www.bhphotovideo.com/c/product/1221495-REG/aputure_v_at_array_trans_wireless_video.html