In order to enable prop_physics_override chairs to be picked up, you'll have to adjust their mass using the Mass Scale field, as they're all too heavy to be lifted by default (a model's associated mass is listed in the info tab when selecting an entity's world model). An object has to weigh 85 kg or less if you want the player to be able to pick it up.

Wheatley: "I'm gonna go with... true... to that one..."

I did find one that shows up if I use it as a static model. chair_padded_office_1980. This will do for my current purpose, but I am curious why some don't seem to show up either as static or physics. In particular I was trying to use: props_motel/armchair.

I agree that it's a bit strange that some of the chair models are static while others are dynamic and only one (as far as I know) is a physics prop. I found it rather disconcerting while playing the game.