Can someone explain the logic of making a "TCAS RA" call to ATC during a TCAS recovery?
In a nutshell, it's so the controller is then aware that the aircraft he/she may have been expecting to do one thing is in fact likely to be doing something else. This might then "save the day" vs other traffic the controller might have been about to clear for a descent/climb into the area of conflict.
(Example in English: If we see an aircraft has descended through level X, another aircraft above can then be cleared to descend to the level just vacated by the one below. If the lower one then starts to climb in response to an RA, there is no separation. Of course, a subsequent RA, or even a series of them, is likely.)
The other big reason, is that if the controller notices an aircraft climbing when it should be descending (for example) and observes a potential loss of separation about to occur, control instructions - perhaps quite agitated control instructions - will continue to be passed.
Making that call should ensure that the controller knows to immediately stop controlling the event. Without that call, the controller will/should keep attempting to provide separation.
(In practice, it takes a bit of conscious will power to "butt out". Goes against all prior training. That's probably part of the reason there are quite a lot of anecdotal reports of "butting-out" not really happening.)