Your understanding of propensity matching is correct: the goal is to obtain matched sets of exposed to unexposed. I'd be curious about the studies where the diseased subset is matched to a healthy subset balancing risk of exposure. Technically, this approach should provide unbiased inference in a case control study where the log odds of disease given exposure is equal to the log odds of exposure given disease.
The problem with analyzing outcome-dependent samples with a Cox model is that the effect measure is not a rate ratio. If controls are matched to cases, then the person-time among the cases is much smaller than expected, and the hazard ratio does not approximate the relative risk in an unconditional sample. A similar issue arises when you analyze matched pairs in a Cox model.
There really is no satisfactory way of analyzing paired survival data. You might consider a Poisson GLM with a random intercept for each pair and a flexible adjustment for the log of the exposure time to estimate a rate ratio to mimic the trait of a proportional hazard and an arbitrary baseline hazard function, although the reduced degrees of freedom are of note.