Hi, sorry for not getting to you earlier, your question is relevant and well written.
This is slightly out of my expertise, but since nobody else responded I’ll give it a try.
I’ll start by noting that there are many ways to code factor predictors for regression. One related (or maybe even identical) to the approach described by Gelman is IMHO “effect coding”, but see also an alternative at Symmetric weakly informative priors for categorical predictors - #2 by jsocolar
If you do effect coding with balanced predictors, then the intercept corresponds to the mean value for the whole population. I don’t think that the rescaling above maintains the interpretation, but I think it could be close.
In any case, I often find it hard to interpret model coefficients on their own and prefer using model predictions for interpretation.
If you have enough data to inform the coefficients well, then I don’t think exact choice of coding should affect your results much. I don’t think this option is particularly bad in any case, but if you find the choice of priors/coding influences your inferences, it is definitely worth investigating.
I think a bit more context would be necessary - what kind of hierarchical model do you have in mind? And priors for what coefficients do you care about?
Also tagging @andrewgelman as he might be the best person to summarise his views :-D
Hope this helps at least a bit!