Is your logic valid? Yes.
Does that mean that your conclusion is correct in the circumstances? Absolutely not.
How family meals and events and funding them works is massively influenced by a) what’s normal for that family and b) the relative financial circumstances for the people at the time.
There are lots of family’s where it’s normal for the parents to pay if they’re involved. For those, it would be totally weird to swap things round. There are others where it’s normal for the birthday person to be treated (financially) by others. For them, swapping it round would also be weird.
When the budgets and the traditions line up, it’s all good, everyone’s comfortable and knows what to expect. When they don’t, things get more complicated. Sounds like you and your fiancé have different ‘normals’ on this one. Doesn’t make one fundamentally right or wrong.
Given it’s clearly a well established pattern that doesn’t have a material impact on you (accepting that confusion and surprise can be got past) then accepting the status quo is probably the way to go.
(and yes, it’s weird getting used to seeing different parents do things differently. Even at basic levels, my mother in law sees it as her role
to host anyone who’s visiting them in their home and thus do all the clearing up after meals. I grew up with the clear expectation that you took your turn in clearing up. No one is right or wrong, but it feels odd).