It is not a "foolish" statement. A blown tire (assuming that you can even correctly assess that you have one during the takeoff run), comes under the heading of something you should not ordinarily high speed abort for, for the reasons you have outlined. That (and many other malfuctions) comes under rejected takeoff decision training, which does receive much emphasis in the sim. The second part of that is if you low speed abort, what are you going to do differently and if you continue the takeoff what are you going to do differently while on the runway? Again, assuming you even know you have a failed tire (of course different aircraft may have different physical cues of tire failure). If you assess the problem in time, not raising the gear to prevent further damage is likely what the QRH specifies.
Now, you're company may make tire failures a sim training item, and more power to you if they have the sim time, but I'm saying that on my personal part, I cannot remember ever being given that scenario in an emergency procedures sim and I'm type rated in four Boeing 7-series airplanes from the 707 to the 747, plus experience in 4 engine military aircraft where sim time was much more available. I got my blown tire training on-the-job, and the failure modes I had never caused any serious concern. As is often the case, specific training and emphasis items will vary by company and indeed by national aviation authority.