Most likely not. I'm not sure how the press pool rules work but there seems to be much less opportunity for follow up grilling as compared to The West Wing's portrayal.
It seems there is a fear of being kicked out for causing too much "disruption." Thus the whining about what was said.
Which of itself is where I get bothered by the repeated attacks by the President on "fake media." You have a room of 30+ people begrudgingly regurgitating the bullsh** you feed them, sprinkled with some easy fact checking on their part (oh which his own staff should have done) to correct it, and then he rails about how they 'did him wrong.'
And Alabama Jack with Mountain Dew Mouth eats up this bullsh**.
*Edit* If I was John Kelly, I'd be looking at how to bring a journalist like Daniel Dale into the office to spot check these things before they get out. I'd also be putting a muzzle on Trump, but that's another rant.
I get the feeling Trump is the worst micro-managing boss ever. Even worse is how he's micro-managing from a golf course. Either stay away and let the adults run the room, or sit next to them, shut up, learn from them, and be presidential.