Tom Tango posts his thoughts on how the pitch challenge system will evolve:

I would STRONGLY expect most challenges to eventually be made by the catcher, the field general, and remove it from the ego-filled mind of the batter and pitcher, but that will take time to prove out. Other than the best batter on each team (Soto, Judge, etc), the catcher should get all the challenges.

TangoTiger.com

That seems reasonable. Other batters will likely challenge egregious, high-leverage, third strike calls, but otherwise they won’t be allowed to waste the challenges when the catchers do challenges well.

One point of this article, and the one at FanGraphs that inspired it, is that challenging skill will be as important to catchers as the framing skill. I might disagree with that a bit. The framing skill is a physical skill. Someone needs to work on subtle glove skills, and some will simply do it better than others. Challenging, however, is pattern recognition, and our brains our great at that. MLB hitters already have fantastic pattern recognizers for pitches. With the challenge system, the catcher’s recognizer gets trained with every success and failure. So they all are going to learn what is challengeable and what is not.

At the same time, the umpires will be developing their pattern recognizers as well. So I suspect as time progresses umpires make fewer mistakes, and so catchers make fewer but more accurate challenges, with most of the variability among catchers coming from the amount of experience behind the plate with the system in place.

It should be fun to see how this plays out.