[AllTheCheese]

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first of all, he definitely has folding equity vs our range.

secondly, paying for a card always involves calling, and never involves betting.

For example, lets say we have AA on a J63 flop with two spades (we have no As). We make a continuation bet, villain with 98 of spades raises our bet, and we call. We just gave him a free card. If we knew for a fact that he had a flush draw, then the correct play barring a few weird circumstances would be to 3bet. Instead, we A) allowed him to make the plus ev (vs our range) play of semi bluff raising a flush draw B)gave him a free turn, and C) are most likely going to give him a free river if the turn isn't a spade.

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Where in this paragraph have you explained how betting is the same as "getting a free card". You've said several times that it is, but without explanation.

Consider this example. Suppose in the hand OP posted, Villain had pocket Kings. Against our hand, AA, Villain's hand is a drawing hand. So suppose we check and Villain bets $7 and we call. Is he "getting a free card" there? What if he has a set? Is that "getting a free card" as well? Is betting in position always getting a free card provided you don't get raised? IMO, your use of "getting a free card" being the same as "not calling" is too general. Given that definition, open-shoving 1000BBs preflop is the same as getting five free-cards.