So my younger brother family shares with me, and sometimes his library stops showing up. Usually I can just click play on his game, and a 'request access' menu pops up, and upon clicking that I instantly get verified. Today however, the same thing happened, except no window shows up to verify. The game just straight up does not open. If I go to the actual game executable for any of the games, it launches just fine (normally this would also trigger the 'request access' message box). It doesn't show me in game on steam through this method, and the steam overlay doesn't work.
I've had the same issue sharing libraries with my roommate. Occasionally it will forget that one of our computers is authorized on the other's account, the only workaround I've found is to log in on the other account and reset it. I've had to do this around a dozen times in the last few years, I haven't found a more convenient solution but it does reliably fix the issue.