How common are mayday calls?
From the point of view of a private pilot, the answer is that mayday calls are not terrifically common. I've certainly never heard one made in anger in a couple of hundred flying hours. 

However your assumptions are correct in parts and wrong in others. First lets define what an emergency requiring a possible mayday call is, which answers your question about when it makes sense. 


  Mayday - A condition of being threatened by serious and/or imminent
  danger and of requiring immediate assistance.


That said, some of your assumption is correct:


  I assume when confronted with an emergency on board, they will first deal with it


Absolutely! Pilots are taught to "Aviate, Navigate, Communicate" in that order. Any emergency while airborne requires you to keep the aircraft flying as a first priority. You'll notice Communicate is last after "knowing where you are and where you're going" (Navigate).


  not waste time contacting any one who would not be able to help anyway


As pointed out in comments, this is an incorrect assumption. Once the pilot has some bandwidth to communicate there are numerable ways ATC could help in an emergency, such as (but not limited to):


Clearing surrounding airspace
Aiding with navigation, such as giving vectors to the nearest runway
Prioritizing landings
Helping you once you're (hopefully) safely on the ground - see below.



  And when they have time for such a call, there's no emergency anymore.


Again, an incorrect assumption. Take the example of an engine failure. The aviate part calls for you to set up for maximum glide ratio, and do some checks to see if you can restart your engine. The navigate part could require you to find a suitable field to make an emergency landing. At this point (depending on your altitude) you might have a number of minutes before touchdown. Here's where you tell ATC the problem ("Mayday mayday mayday. G-ABCD, engine failure, 2 miles east of [city], 2,000 ft heading 180 degrees. emergency landing on field beside [big road]"). This has the benefit that they can communicate with emergency services - or at very least send someone out in a car to help you.

As you can see from the above example. The emergency still persists once the pilot has time to tell ATC about it.