Looks like in section 3 when you click the fan "on" in homekit it is sending a setlevel(100) and also an on command. The "on" tries to set low speed but the level 100 tries to set it to high.
So the issue might lie in Homekit.
I will try and test this on my fan I have setup through Homebridge, see what HK sends back to HE for commands. I do not usually use HK to control the fan, just to sometimes see if they are left on and then turn them off.
