While it is normal procedure to expect the PNF (let's call him the co-pilot for sake of another term) to expeditiously advise the captain of an abnormality during the take off run, some captains would have strong reservations of him calling "Stop - Max Brakes" or whatever.
CRM is lovely of course but there are critical areas where it is the captain who wears the consequences of what may be the most vital action in his career -a high speed abort. Lawyers would have a field day if the captain pleaded he was only doing what his subordinate had ordered him to do. Surely the captain is capable of deciding and doesn't need the help of a committee of one. Call the event by all means but there is no need to direct the actions of the captain.
Yes I know there are wonderfully switched on highly experienced copilots and so on. There are also some less switched on inexperienced copilots who could not be trusted to land a Tiger Moth on a 10,000 ft runway. I remember recently asking a chief pilot of a very small British "airline" who operates Boeing 737-200's why Jet Upset (Unusual Attitudes) were not part of the company training syllabus. His answer? He said his copilots (some had less than 500 hours total time) were the best trained in UK and he could not possibly envisage them allowing the aircraft to get into a UA in the first place...Er - Yes?
Back to the tyre burst question. There is no correct answer. Historically it is better to take your chances and keep on going if you are within 20 knots of V1. Controllability is rarely a problem in such events but an abort with less than maximum brakes (tyres) is fraught with danger on limiting runways. Reading of past accidents of this nature is also good insurance against making the same mistake. Isn't there an old saying about learning from the mistakes of others?