Being sleep deprived is hell. It's even recognised as a mode of torture.
Of course we're all grateful to have our kids, we're even more grateful that they are healthy and not sleeping in an hospital bed, of course, but this has nothing to do with anything. You cannot appreciate your life and your little ones when you are really sleep deprived.
No, letting your kids cry themselves to sleep is not the answer for everyone. Some babies sleep through the night from a couple of weeks old, others need a year, it's pure luck. After reading a lot about sleep training, I personally think it's awful to let them cry, but each to their own.
OP< I do feel your pain. My only way to cope is to grab child, put in my bed and go back to sleep (and repeat, the older ones come by themselves). If fussing whilst in my bed, I put back in theirs, close door, let them cry for 2 minutes, pick them again and tell them to be quiet in my bed. It works.