I can only speak for myself, but when I say I count "all the time", I don't mean that I am doing it continuously, but that it features in my dancing very regularly.
I agree that the way to respond to the pulse in foxtrot is a key part of the character of the dance, but many dancers of modest ability struggle to find the right rhythmic expression, although as Alex Moore says, it can be learned, even if it is then contrived. It is not for nothing that foxtrot is usually reckoned as the hardest of the ballroom dances to master, but I think it is also the most rewarding.
I seriously doubt that the rhythm of the dance can be expressed correctly unless you can first hear the music's underlying pulse with complete clarity, and then have sufficient dance experience and technique to appear to be dancing with the unhurried and smooth movement that characterises the dance. Never on the beat, but always in touch with it.