You’re paying twice because the fees pay for the delivery service/infrastructure, while the ads pay for the content. They are made by different companies. You might as well ask why you have to pay for Netflix when you’re already paying your ISP.
That article is actually quite odd; the then-existing cable-only channels it mentions were mostly also commercially supported from day one, the ones they report on considering commercial models are channels preparing to go online, not then-operational ad-free channels. While they point to a public expectation of ad-free cable, they don't point to any actual history outside of premium channels (that generally remain ad-free), and other than public access channels, few non-premium cable channels were ever ad-free (except PBS broadcast channels carried by cable operators, when PBS had public funding that allowed it to operate ad free, but that had nothing to do with cable.)