I haven't really thought it through too much but fairly small - smaller than the HPG Kit Bag, I think, but similar: I would want it fairly flat, basically there to hold all the critical stuff from my pockets, but when I'm either carrying my kid on my shoulders, or running around in summer gear which is board shorts and t-shirts.
But my pockets are generally pretty freaking full on account of my survival-oriented semi-suburban off-grid wilderness ocean dad lifestyle.
Now that I've mentioned the kit bag twice I've gone back to look at the HPG site and I'm feeling a bit of interest in this Snubby Belt Pack:
https://hillpeoplegear.com/Products/.../ProductID/129
That's roughly the proportions (and actually the concept) I was thinking of, I guess.
I can't get the Aer site to load correctly and the first picture I looked at of it the Day Sling 2 made me think it was more shoebox- or tube-proportioned for some reason, and I initially discounted it, but I just went and looked at a third party review and...yeah, that looks REALLY similar to the HPG Snubby Belt Pack, actually. I think my application is similar to what Rob is describing, also, so I can see how similar products would appeal.