I don't think that you should be able to tell where the seam is, nevermind the paper tape. Paint won't fix it. He just figures by the time you find this out he will have been paid and by shystering the next job. One pretty cool way to call his bluff is to go out and buy your primer now, a really good one with lots of pigment - like bin or kilz, and put a coat or two on at least the areas with the suspect seams. Then he can see for himself that paint won't fix it. This stuff dries very fast so you could tell in about 1/2 hour. He is full of it.
As for the calking, If your trim (baseboard) is painted (as opposed to natural finish) then there isn't anything wrong with using a little calking to fill in a small gap here and there. This is pretty normal practice. If done normally and correctly then it is usually done AFTER the primer coat, not to fill large gaps or holes and done with paintable calking. If done correctly, then you should not be able to tell it is there after it has been painted. Again, if you are in charge of the painting then throw some primer on it and see if you still notice it. If you still see a depression where the crack was that he was trying to fill - after painting it, then the void may have been too large to calk and he should have mitered or scribed the trim to fit the area better. If the gap was 1/8" or larger then it was too big to try to calk.