A guy I've been on dates with would not let me pay. Should I be worried?

I've been on 2 dates with this guy, both times we did activities then a lunch or dinner.

On first day when the check came for the dinner, I asked if I should get it and he paid for the games that we played earlier, he said he got it but said he appreciated it.

On second date when we were at check-out for the activities we purchased, I put my credit card to the card machine and said I should get this one since he paid for everything. He calmly said "Do you mind?" I thought he meant if I minded paying, but found out he meant if I minded that he paid. I was going to swipe my card but realized it was a touchless payment, and that his card went through when he asked if I minded.

When the check came at lunch, I put my credit card next to it and said I should get this one, since I didn't want him to pay for everything. He said just take him out next time, put his card and removed mine.

Later on I texted him what he meant by taking him out and if he wanted me to plan for next date (he's been doing the planning and told me not to worry). He said he only said that, and it was his pleasure to treat me to a good time and he's got it [the check].

I've enjoyed conversations with him & some deep ones, and would go on dates with him, but I'm feeling a little stressed šŸ˜… I've heard that a woman said she didn't want guys to pay as a way to say that she had to go on dates with them.

I agree, but it doesn't seem like that's his intention (maybe not?). I'm also worried if he has expectations of me that's beyond the present moments (dates). For example, because he pays for dates, I'd have to assume traditional female roles, such as cooking, do the dishes, cleaning, etc.

He knows that I have a job & shown interest in learning about what I do, and he doesn't expect his partner to take care of kids because he said he leaves the decision of having children to the woman. (We're not in a relationship yet.)

Should I be worried (in addition to feeling bad) that he's been paying?
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We have 2 dates planned and coming up. One is a dessert date, which I'm not too worried, but wonder if he would at least let me pay this one! Also don't want to annoy him by trying to get every check (but what should I do? Let him pay? 😅 Fight for the check?) One is activities dinner. When I saw it was a $100 restaurant I felt a little pressured, but found that the food that I would order are around $25 per dish, so felt better, but still, what do I?
A guy I've been on dates with would not let me pay. Should I be worried?
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