I think it's absolutely clear that for some people in some conditions, pain would not only last them until their deaths, but would keep getting worse.
Finding silver linings is a separate issue and people's ability to do that and adapt to the pain differs and any generalization in that sense is bound to be fallacious.
That's a generalization that sounds poetic but I don't think it has been shown to always be the case. For some people the threshold does not adjust with time and the pain is not sharp enough to get you to pass out.
Our minds and bodies are not machines and don't always act according to the pattern you are describing and there are always things that could malfunction to prevent the mechanisms that you are describing from working the way you are describing.
Additionally, what good is passing out or sleeping if you are still in pain after you wake up. Even if the pain is not constant for some and they do get a breaks, it can still be unbearable to them.