From a PT JUNKIE
Befor I begin let it be clear that I AM NOT A PT GOD! I am a PT junkie there is a difference.
Training while tired I agree to a certain point. Mentally you are right. Strong mind strong body. But there is a time when you do have to rest to allow the body to recouperate from the vigors of training. Rest as I have found out is a weapon. The body goes through a process of internal communications if you will that will help you in more ways than one. Allow me to explain.
During periods of intense exercise for prolong periods of time, the body communicates to each other on the same level ie, lungs, pulse, heart rate, mind, ect. This is done to allow you to perform at your best. The mental aspect of it allows you make your body work passed its comfort levels. Remember strong mind, strong body? This is where it comes in. Working passed your comfort levels, definately will strengthen your mind, as well as body.
Every time you go pass your comfort levels you become stronger both mentally and physically.
Now be very careful, you may "crash" meaning you have hit a plateau when your body saids enough is enough and forces you to rest. This is often done by severe depletion of certain nutrients in your system like Glycogen levels a very important nutrient within your system that is used for energy.
Overall, training while your tired is good to a point, however you must rest in order to allow your body and mind to be able to handle the next training session. Think of rest as an essential nutrient to the body and you will see how you are able to come back stronger and better than before.
"I succeed not because I want to, but because you expect me to fail"