So if psychological exhaustion is something that naturally happens, selection could push it off until physical exhaustion so that you can keep contributing to the tribe as long

But why would psychological exhaustion naturally happen at a rate that's fast enough to be relevant? There's no second law of thermodynamics for algorithms; it's simpler for evolution to build a brain that never gets psychologically exhausted, so that's (to a first approximation) what would happen. It seems that evolution layered a routine for suicide on top of our brains too, but it seems that that routine doesn't check for "how old are you", it checks for "how low status are you", probably because your family may lose resources trying to help you and thereby reduce the inclusive genetic fitness of your genes etc.

The argument that you're making (https://en.wikipedia.org/wiki/Selection_shadow) specifically only works for things that need active effort to prevent them from breaking, which tends to mean physical stuff. Psychology isn't really susceptible in the same way, because although or psychological health will be in the selection shadow at e.g. age 300, there isn't that pressure of thermodynamics to break it.