Well, a chlorinator does allow you to control the speed at which you are introducing the chlorine, the skimmer isn't controllable.
We don't use tabs, we do BBB, so I cannot tell you from experience what can happen, but I've heard from others with pools that chlorine does in fact make the skimmer and basket brittle, a brittle part can still work, it all depends on how well you take care of stuff and how much abuse it sees, another problem with the puck in the skimmer is once the system is shut off, the puck keeps melting and you end up with a high concentration of Chlorine in that spot, so your liner is also affected in that area, it will become brittle early and could become a potential problem spot.
I'm not saying you can't do it, i'm recommending on using a 10 dollar floatie to save your 50 dollar skimmer, maybe the skimmer will last 15 years, maybe 25, but maybe it COULD have lasted 50 - i'll get back to you on how my pools doing in 30 or so years through taking care of it.
And i still advise against chlorinators or even the use of pucks unless it's needed.