Everything has to be paid for. I would expect a cheaper job if I took the old kitchen out ready for the new one to be fitted. Similarly tradesmen have to pay to dump stuff at a tip whereas private householders can dump most things free.
Not quite the same but we had our bathroom redone a couple of weeks ago. The builders ripped out the old one, ordered a skip and put all the stuff out of the bathroom in it. You really should have discussed this with your fitters.