I've started to have an issue with my Specialized Air Tool Comp as well, and I'm also suspecting wear to the pump head gasket. It works fine on Schraeder valves but I really have to work to get it aligned as precisely as it wants to be on Presta valves. I just looked it up - I wasn't really expecting a cheap pump to be rebuildable, but in fact Spec' has a rebuild kit for $3 on their website. Yay! Too bad the nearest Spec' dealers are so far from me.

I'm thinking the OP has Presta valves as well, since Schraeder valve stems don't normally bend. So there wouldn't be an issue with the pin depressing the valve core, because Presta valves aren't spring-loaded; the air pressure in the tire seals the valve, and when the air pressure in the pump head exceeds the air pressure in the tire, it opens by itself.

My guess with my pump is that gasket wear means the valve stem gets easily misaligned inside the pump head, so the nut on top of the valve core contacts the inside of the gasket, which both prevents it from opening and risks bending. If your pump has been fine for a while and just started exhibiting these symptoms, I'd guess it's the same for yours.

I'm also guessing that if I hadn't been using the pump for both Schraeder and Presta valves it wouldn't have worn out, or at least not nearly so soon. For all they say the pump's supposed to be good for both, it's obviously a really tight fit over the metal threads of a Schraeder valve, and quick wear on the gasket. So maybe I should just get a new pump and dedicate the old one to the Schraeder valves...