I asked this question to a University of North Carolina professor and got the following response. 

Remember that propensity score analysis, correctly performed, is supposed to replicate a randomized experiment.  Suppose that there is a clinical trial where 40% of those under 50 are given drug A while 60% receive a placebo.  Similarly, 60% of those above age 50 are given a placebo and 40% receive drug A.  Once somebody has entered the clinical trial there is sense in saying that they either received the treatment or did not. But before they entered the clinical trial there is a real sense in saying that those under 50 had a 60% chance of receiving the placebo.

Similarly, in an observational study, if it's discovered that 60% of those under 50 chose not to undergo therapy while 40% chose drug A one can say that before taking the drug there was a 40% chance someone under 50 used drug A.  (Of course, this is a deliberately simplified example with only one confounder but the simplicity is deliberate to focus on the point.)  After the experiment has begun and patient 1 has started therapy on drug A it makes no sense to say that there is a probability they are using drug A: either she is or isn't.  But before the choice of therapy there is a meaningful sense of saying that patient 1 had a certain probability of using treatment A.  

The difficulty is solved then by not confusing pre-experiment randomization/probability with statements about whether the participants are receiving the treatment after the experiment has already begun. Obviously, nobody would say that randomization is not meaningful in a double blind clinical trial.  By analogy, the same terminology can be applied to propensity score analysis.