Taking the battery as an example. Intuitively, I understand that without the chemical forces that continuously maintains a voltage difference between the two terminals, the current will stop because, eventually, the two terminals of the battery will obtain the same voltage.
However, most textbooks I have read emphasize that the source of an EMF must be non-electrostatic (or non-conservative) in origin. From my understanding, as stated above, I don't see a connection between the chemical forces that maintain a voltage difference between two battery terminals and that they must be non-electrostatic (or non-conservative). It seems to me that the mechanism of maintaining a voltage difference (in this case, the chemical forces) still works just fine without the requirement of the forces being non-conservative.
My question is: Why is the fact that the source of an EMF being non-conservative important? I'd prefer an explanation in the context of my understanding of EMF, which is stated in the first paragraph. Also, please limit the explanation to classical EM only.
If my understanding is missing something, which might have rendered my question to appear "nonsense", please help me correct my misconception(s).