So if the textile mills in the North which used 100% Southern cotton were forced out of business by cheeper imports from England and France who used maybe 90% Southern cotton, demand for Souther cotton would be higher?
That makes no sense at all which is likely why no one made that argument in 1860.
The tariff on imported cotton finished goods raised the price of such goods to all domestic consumers, lowering the quantity demanded below what it would have been otherwise. If the demand for a finished (or “final”) good falls, the demand for the specialized inputs - in this case, raw cotton - used to produce that finished good falls as well.
The sense of injustice expressed by Southern politicians of that era reflected the fact that Southern suppliers of raw materials participated in competitive world markets, while Southern demanders of imported manufactured goods faced artificially high prices because of the tariffs enacted, in part, to protect Northern industries.