Why are batteries modeled as ideal voltage sources?

When using batteries as your source in a physics/engineering problem, it’s considered as a voltage source. You’d say “a 9V battery”. However, looking at the battery as an electrochemical cell, I’d guess that it’s a current source. The reaction is going this fast and therefore provides this rate of electrons to the load. The only problem with me looking at it that way, is that I can also see it being a voltage source anyway: the ions diffuse through the electrolyte so fast that the total voltage across the battery is pretty much constant.

I still haven’t fully convinced myself that this chemistry based argument for being a voltage source really makes sense, but the question still remains: why are batteries modeled as voltage sources?