"Anonymous. I'm gathering from your writting that you are one of those who thinks they can dance to the melody"
No, one dances to the pattern of the rhythm - but not to its invidual details.
"and don't have to count the beats."
Any music teacher will strongly advise you against counting the beats, except when initially figuring something out. Counting is almost guaranteed to put you off time, and impede you ability to respond to expressive changes by the other musicians. And as a dancer, you are one of the musicians - merely playing your body as your instrument, so the same cautions absolutely apply.
"I've had a lot of lessons from a lot of people over a long period of time. I have never met one who didn't count the timing.In our classes anyone of us can be called upon to dance to music and count the beats."
You need to know it, yes, because it is an important skill for puzzling out new things. But it is incompatible with dancing artistically - at some point you have to switch over to dancing the trends, otherwise you will never by anything more than a robot lagging pathetically behind the beat. To actually be on time, you have to generate your internal timing, and synchronize it's major features (barlines approximately) to the music. If you try to match the beats, you are guaranteed to be off time.
You must outgrow depending on counting for actually dancing, but you will never outgrow making use of it as a learning tool when attacking a challenging problem.