Maybe what Picard experience was Kirk in a fake reality, too. Reasoning: Kirk is still wearing a Starfleet-like under-uniform outfit, suggesting he never really left it and its always in hte back of his mind, and that his mind knew things were not right so he created a fake reality for him to always keep him off kilter and hopefully one day wake up to where he was.
As with Picard and Guinan, those are the clothes he was wearing when he went in. Yes, the implication is that he wears the vest under the jacket, each with its own badge (you can see the outer jacket, still with its delta shield, draped over a nearby tree trunk while Kirk is chopping wood). They did the same thing with Sisko in DS9.

) Nicholas Meyer and Denny Martin Flinn actually took some chances with the TOS characters and had them do some surprising unexpected things in their last movie. Dillard seemed more preoccupied with sanding off all the edges than actually adapting the movie's story.