Using binary outcome variables in real-world data studies must be wrong? Please be gentle if it's a stupid(ly easy) question:
In medical literature lot's of randomised clinical trials use binary outcome variables, such as 90% reduction in Y, or Y<(a certain threshold). Aside from issues of power, I'm okay with this since all groups have the same starting point (baseline value).
These binary outcomes are often used in observational, real-world studies, where the starting point for each comparison groups is clearly different. 90% reduction is clearly different for a group starting at 100, than a groups starting at 10 (absolute reduction ten-fold different).
Intuitively, I struggle to believe that "adjusting for the baseline value" or propensity scores solve the problem in these observational studies. Since, as in the above extreme example, there is no overlap in distribution of baseline values in these two groups. How can you "hold the baseline value constant" as is claimed in regression adjustment?
 A: Your problem doesn't seem to be with binary outcomes or with propensity score analysis but rather with "percentage change from baseline" as the outcome.
Note that "raw change from baseline" is not problematic. If two groups are equated (i.e., with propensity scores or regression) to have the same average baseline, then the difference in change scores is equal to the difference in raw outcomes. If it's impossible to equate the two groups without extrapolation because of a lack of overlap, then confounding becomes an issue, but that is a problem in all observational studies and not just ones that use raw change as an outcome. If your problem is that for those with low baselines, there is little room to decrease, then you are describing a nonlinearity or floor effect that needs to be modeled, which can easily be done using a generalized linear model or a flexible linear model. For example, a model that contained an asymptote at 0 (like a fractional logistic regression) would reflect the fact that there is little decrease possible for someone already at a very low level of the outcome at baseline.
Percentage change can be problematic, as you have described. Two groups with the same raw change may have different percentage change if they start at different baselines, and adjusting for this by balancing on average baselines will not solve that problem. Even if the two groups had the same average baseline and the treatment was totally ineffective, you might see different percentage changes just based on the fact that those in the tails of the baseline distribution have different potentials for change. However, if you could exactly balance the entire distribution of the baseline, then the difference in the percentage change would be valid and  not confounded by the baseline. This goes way farther than most propensity score analyses go, though, but it is possible. For example, finding exact matches on the baseline measure would ensure that the distributions of the baseline in both groups were exactly balanced, so a simple difference in either the percentage change or absolute change would not be affected by confounding due to differing baseline levels.
