Thanks, all information that I can see is the same as in my case.
I must admit that I have missed the part about selecting cloud connection explicitly. I always thought automatic should use any available method… 
Anyway, it works now.
However, there is no motor speed control in Manual/Favorite mode. I tried to manually set the Custom service.favorite level or Custom service.moto speed rpm attributes of the created fan entity, but it appears these are read-only.
I also tried calling a service below, using parameters from https://home.miot-spec.com/spec?type=urn:miot-spec-v2:device:air-purifier:0000A007:zhimi-cpa4:1 but it does not work (not sure if I got the command right, probably not)
service: xiaomi_miot.set_property
data:
entity_id: fan.zhimi_cpa4_21de_air_purifier_2
field: custom-service.favorite-level
value: 10
Air Purifier 4C(zhimi.airp.cpa4): Set miot property custom-service.favorite-level(10) failed: property not found