Thanks for reading my 1997 JASA paper!
If I had a do-over, I would rephrase my comment that a (single-step) Bonferroni adjusted p-value is not a probability "per se." (And I would no longer use the dreaded "per se." Yecchhhh!)
The Bonferroni adjusted p is in fact an upper bound on the probability that the smallest (random) p-value is smaller than (smaller than or equal to in the discrete case) the given (fixed) p-value, assuming the complete null model describes the randomness. And certainly, 1.0 is an upper bound on any probability.
But the bigger and more important point of my paper is that you can find these adjusted p-values exactly in such a way that accounts for the correlations between the multiple test statistics, assuming the classical linear model. These exact adjusted p-values are in fact probabilities when calculated in single-step fashion; see p. 302 of my JASA paper for the math. (To get the single-step p-values, you need to modify the expression somewhat; see my 1993 Wiley-Interscience book and my SAS book). While I used an enhanced Monte Carlo method to approximate this exact probability, better methods have been developed since; please see Hothorn, T., Bretz, F., and Westfall, P. (2008). Simultaneous Inference in General Parametric Models, Biometrical Journal 50(3), 346–363.
So, single-step adjusted p-values, when computed exactly, are bona fide probabilities.
But, except for the smallest one, step-down adjusted p-values are not bona fide probabilities. They are constructed from bona fide probabilities, but they are not probabilities.
Hope this helps!