Simple concept here. Tire blown or not by the time you recognize and react when you are V1 -5kts, you more than likely have already exceeded V1. Other concerns may be if you abort the braking efectiveness is reduced due to the blown tire (tyre), however the wing at that speed is producing enough lift to get you safely airborne. If all engines norm, you can leave the gear down for the circut.
Landing at a decreasing airspeed with all the runway infront of you with ARFF standing by makes a lot more sense than trying to abort with decreased braking and minimal runway in front of you.
Don't get to caught up in abnormals on T/O. Loss of control, ENG Failure, fire-smoke in A/C are couple to really consider. But most abnormals can standby for a circut and landing.
Bottom line...if the Aircraft can safely make another circut and landing, don't abort near V1 .