A few things to consider here - and just keep in mind that my knowledge on this subject comes from studying the major races to understand why people won/lost and communicating with drivers and/or mechanics after the race. I also read a lot of race reports as well.
A tire strategy for a one hour race is a big deal. Some drivers decide not to change tires while others do. Those that don't change their tires often finish with very little material left on the wheel. It's a very calculated risk on their part. Many drivers have won or lost by not changing tires. Is there a risk of running down to the plastic? Yes. As has been stated, there is a risk to changing tires as well, including flameouts (as Brian mentioned), the mechanic putting the wrong wheel on a corner, stripping threads, the car being dropped and damaged, and so forth. So no matter what a driver decides, there is risk.
Because the car drives and performs differently with different tire sizes, it's important to consider the implications at the beginning, middle and end of the race. When are you faster and when are you slower? When does handling get sloppy? When do you have no choice but to change the tires, even if your strategy was not to change tires? How much performance degradation can you handle without wrecking the car? If you do plan on changing tires, how much of a lead can you establish so that you can still be on the lead lap (or ahead of it) when you get back out on the track? If everyone is on the lead lap and you need to change tires – and you lose position, how many laps is it going to take for you to get back on the lead lap – and is that even possible given the competition? When you change your tires is also important. So as you can see, as mentioned earlier - this is a big deal.
As the driver, you need to consult with your pit man and plan ahead for various contingencies. You're essentially calling the shots and the risk is all yours. Your pit man will have an opinion of course, but ultimately, he should support your decision – regardless of if he agrees with you or not. The pit man's experience counts for something, but you're the one driving - so don't blame your pit man for accepting his advice if things go wrong.
As a side note, some drivers do have different shore and different size tires left-to-right, depending on how the tires wear. Figuring that out comes with experience.
Also keep in mind that pitting for fuel eats time. You can be the fastest guy on the track, but if your pit stops are slow or inconsistent, you can lose lots of time to guys that are just a bit slower out on the track.
Hope this helps.