Insisting the other person pay has a lot more signalling implications than you think. Depending on the rest of your personality/appearance, it can make you seem like a cheapskate, or like someone who's overconfident, or someone who gets laid a lot so doesn't care about impressing random women, etc.

You're also viewing attraction in a very weird way. Most romance isn't love at first sight. Making a good impression on someone and seeing them a lot increases the amount they like you, and it is very unlikely that you meet someone for a first date and they totally want to be in a relationship right away, regardless of bill splitting. It's far more likely that they have some impression of you, either positive or negative, that bill-splitting will influence one way or the other depending on their personality and yours.