Here are my 2 cents. The great majority of people take the boards right after residency graduation, whether they join a practice or continue on in fellowship. There are some advantages to this: what you learned in pediatric residency is most fresh at this time (and that actually came in handy during the exam, I would go so far as to say that about 1/2 of the questions I felt confident about were not because I happened the cram that material before the exam but bc of my training/what we did in residency). Also, this also seems to be what most jobs and in particular fellowship programs expect; there are in fact some fellowships that will impose academic remediation/even termination you fail the exam! Also, most peds fellowships are structured such that you do a lot of your research PGY5 and 6; you certainly don't want board studying from preventing you to do your research or worse, hindering your clinical development as a sub-specialist. And also, I really do think for studying for these kinds of exams we really probably do have enough time; it's just that because it's such a big deal and we freak out about it that it seems we don't. I would say that I was fairly busy during my first half of 1st year fellowship and was even asked to write a book chapter during my studying time and I still found time to hammer out the details and pass.
On the other hand, there are definitely people who would probably benefit from delaying the exam a year, whether this be to not enough preparation throughout/during residency, a weak knowledge base/poor test taking skills entering residency. There are people that I know that took they exam after residency b/c that's what everyone did and failed and probably should not have taken it then. That being said, if someone has a poor history of taking tests, I am not sure how much 1 year of studying would improve things.
So, I think the moral of the story is to start thinking about the boards early in residency, particularly if someone has a history of poor test taking performance. This doesn't mean start reading through medstudy books or spending 30min a day doing PREP questions intern year but probably at least doing 10-20 questions or so every month on average and if you know you are weak on a particular subject, to at least read about a few of those a year. If by mid-year 2 you are feeling like you would likely fail (ie scored in the bottom 20% for ITE for PGY1, PGY2), then I would seriously consider delaying the exam but otherwise if you are on the lower end, that might involve putting in some more time during PGY2 and PGY3 before the "final push" from July-Oct before the exam. It's certainly a better use of time and money to take it when you are ready and hopefully employers and fellowship programs would understand.