Running full speed is not for endurance.
Endurance training is exercising "LESS THAN" 70-80% of your full speed for a longer and longer period of time as you continue training. Endurance determines the length of time you can do any given exercise, not how far you can push the exercise.
People can't push past about 70 to 80% of their "ultimate strength" for more than a short time no matter what their endurance. Those people running forever have pushed their strength limit high enough so that running at that pace is below their 70-80%. If they push beyond 80% of THEIR limit, they will tire out in a few minutes, too. Their 80% level just happens to be higher than running at that pace.
You will eventually increase your strength limit, which periodically running at full speed can help with, but you will need to make most of your exercise happen below that 80% mark so you can build endurance.
About that ten minutes... I believe that ten minutes is just a nice target for a beginner and is probably achieveable in a few weeks or less. Once you reach ten, you will be pushing for fifteen, then twenty, and so on.