Does Hospital Debts incur the 2 experience point penalty if the investigator is defeated?
This was fixed in the Official FAQ Document:


  Rules Reference page 10, column 1, “Elimination”
  Add the following, as step 0 (i.e. before step 1): “0. For the purpose of resolving weakness cards, the game has ended for the eliminated investigator. Trigger any “when the game ends” abilities on each weakness the eliminated investigator owns that is in play.  Then, remove those weaknesses from the game.”