If a slime tube hardens when exposed to air, why doesn't it harden when I pump it?
For the same reason a salt-water solution in a closed half-filled bottle will not evaporate - even though it is in contact with some air.

Open the lid - and eventually the water evaporates to leave just salt.

The slime solution is in a "balance" with the air in the tube - since it is a "closed" system. When the tube is punctured the slime flowing out of the puncture is no longer in a "closed" system and evaporates / dries in the open air.

Takes me back to Chemistry and reaction kinetics (yuck) .. but anyway that's my take on it.