So, one of you is saying that the spacing between the lines of latitude are consistent, and the other is saying that they are not consistent. Which is it?
We are both saying that latitude is consistent on a round earth, but not a flat one.
Both of you claim that the lines of latitude don't work when flattened to a plane - but there are flat maps with latitude and longitude on them available in every convenience store. It's a coordinate system, nothing more.
First of all, you do understand that the "flat maps" that you are referring to are flat projections of a round earth, don't you? Projections always introduce some sort of distortion on one axis or another.
Secondly, it's not so much that sighting Polaris can't work on flat earth, it's just that the results are not what have been observed for countless years.