To the best of my knowledge the military does not train anyone in handling no win senarios. Nor do they spend any time teaching that there is no such thing as a...
At its best its a literary device, and a not particularly good one. Kirk, in this case, may actually be the right one here: if you are in a 'no win situation' you aren't trying hard enough to win. The only time you really REALLY are in a no-win situation, in real life, is when you've given up and stop trying.
History has demonstrated this time and again. Marathon and Thermapolae both come to mind. That dude that knocked out Mike Tyson? yeah, him too.
In order to create a 'no win' situation it is necessary to artificially restrict the options for the person in that situation. Rules, if you will.
Bujold has an interesting one in one of her Miles Vorkosigan books, where in his military training he's given a situation involving toxic gas and a malfunctioning emergency oxygen system. Only one member of the crew could possibly survive, and then barely. With his simple pocket knife (equivilent) he splices IV tubing from the medical kit and provides enough oxygen for everyone, 'winning' his kobiyashi maru quite handily.
Kirk, of course, is dealing with a computer program senario, and as anyone whose played video games regularly can attest, the programmers ALWAYS cheat for the computer. Kirk can't win not because he's outnumbered but because 'chance' has been utterly subverted by the program to make sure everything 'possible' in the program will fail. Kirk, of course, wins by hacking the program.
Fatalism is not something to cultivate in the military, I think. It can serve its purposes in the rank and file (See also:Bushido), but when applied to officers it creates an appalling lack of creativity and strategic thought, the tendancy to look at problems and just 'give up' rather than seek solutions. Its bad for morale in general. Spock's 'lesson' is quite... illogical.