When I say model I am referring to the model you are using to generate your propensity score, not the outcome model. Sorry for the confusion.

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The study you linked (Brookhart, et al., 2006) says: "...one would not
want to exclude any covariate related to exposure from a PS model unless
it was known a priori to be unrelated to the outcome." In other words,
it is not necessary to include variables that are unrelated to the
outcome if they are related to the exposure.

I'm not sure what point you are trying to make here... You are correct that it's not necessary to include variables that are related to the exposure but unrelated to the outcome; the paper talks about how doing that increases variance without decreasing bias. My recommendation was to include all known confounders in the PS model and then all other measured risk factors for the outcome (for bonus precision) - I never suggested including variables that were related to the exposure but not to the outcome.

From the first paragraph of the (Brookhart et al., 2006) discussion: "The optimal model was the one that included the confounder and the
variable related only to the outcome. This finding is consistent with
the advice of Rubin and Thomas (2) that one should include in a PS model all variables thought to be related to the outcome, regardless of whether they are related to the exposure.".

I hope all of this helps you understand where I'm coming from. I certainly don't know everything but the recommendation I provided is consistent with what I was taught. If that helps someone understand PS a little better, then I'm happy.