You guys aren't alone. We fly stupidly tired in the military too, especially overseas where the "mission" pushes the more flexible rules aside.
For instance, when we sit Bravo alert, we're on call for 24 hours (sounds like you guys). At 23hrs and 59min, you can be alerted for a planned 16-hour mission that can extend to 18-hour with PIC authority. So if you had *just* gotten to sleep when that call came, you're talking about potentially shooting an ILS approach to mins after an 18-hour day on possibly zero sleep in the last 28-32 hours.
And yes, that does happen. Luckily, the Air Force culture is (IMO) VERY safety conscious, and the one or two times I called uncle on that exact scenario never came back to me negatively. I would be asked by the DO "do you feel safe to fly?", I'd answer "no", and that'd be the end of it.
Eventually, enough crews were waving the • flag that HQ stopped intentionally launching alerts without giving the crews a heads-up (unless it was a true emergency), and/or the squadrons started putting two crews on each alert cycle (one day crew, one night crew).
I heartily agree with all of you- it may take a crash of a legally flying crew to raise the issue.