There's countersignaling at play too. A woman has very little information about your true "value" at first blush. If you buy drinks for her, that indicates that you think she must be a good catch, i.e. she's out of your league. Knowing nothing about you other than that you think she's out of your league, why should she go out with you?
So guys "play it cool" as a way of signaling that their true social value is well above the woman's, and they don't need to play her with gifts for them to be worth paying attention to. It's a way of saying "I'm so cool that I don't need to buy you drinks to make you pay attention to me."
That's why dating is usually described as push/pull. You want to signal your interest and commitment to a woman. However, if you signal too hard, then the impression she gets is that you have nothing to offer other than your undying love (hence "coming on too strong"). But if you signal too little, it's indistinguishable from someone who doesn't care about her at all.