Split AC systems remove moisture from the air when they’re running too, that’s why there’s a condensate drain on the evaporator (and the condenser, but that is outside).
A condensate dehumidifier is just an air conditioner where the condenser is inside the same space as the evaporator. It uses vapor compression refrigeration.
Granted, the condenser and evaporator both condensing moisture inside of the space does remove more moisture so you’re correct about it being more effective, but an A/C unit does a pretty good job where I live, and cooler air holds less moisture.
I never deal with cold and damp conditions, a dehumidifier probably works well in the UK. I’ve lived in forced air and radiant heated buildings and they both dry out the air like crazy in the winter so dehumidification isn’t needed, humidification is.
It's probably relative to how much humidity there is where you are. I'm in a very humid area, and I have humidity sensors. Leaving an AC unit on dry mode in a small room does basically nothing - at most it will shift humidity by a few percentage points. Meanwhile if I use the actual dehumidifier I can drop from 85-90% down to 65%. If I turn the machine off for a few hours (or move it to another room) it'll be back up in no time.