Self-hosted Speed Test for HTML5 and more. Easy setup, examples, configurable, mobile friendly. Supports PHP, Node, Multiple servers, and more - librespeed/speedtest
My guess that websocket add additional overhead both in size (header) and complexity as browser and server need to encode/decode it making it more CPU intensive not to mention harder to implement.
This server is just a piece of software that accepts and responds to TCP sockets. It could be anything, but you're not consuming more CPU power by doing less work.
The http protocol requires you prepend every message with a bunch of headers and for them to have a static size. (tho plenty of apps allow you to bend the rules quite a bit)
After switching to a websocket, none of that applies. You can just dump data straight into the TCP socket. Or TLS stream that goes into the TCP socket. But that would be same for the http requests
On the server side to send you data, using any web server with mmap support will probably be less CPU intensive than app that handles websocket, but yes, the details matter as when reading a lot of small files vs websocket, then websocket could be better for CPU usage especially when you could generate data.
But once again using plain old http allow you to use the speestest software against any CDN very easy IMO.