I think the point is to teach people that apparently obvious things are not always actually true. This is extremely important in math as you move on, as i'm sure you know. Many "obviously true" things are not actually true when you get really serious about formalization. Teaching kids that "seems true to me" isn't good enough is a critical step along that path.

Exactly. You can't teach people that sometimes their intuition is wrong if they don't have any intuition to begin with.

I see this in theory, but I'm having a hard time thinking of geometric problems that violate intuition and that are appropriate to teach people who have no calculus and limited algebraic skills (admittedly this limits statistical problems too). I'm actually curious if you have any good examples, because I don't.

And I think that's where my objection comes in: if you spend the entire first year just developing that intuition but forcing a rote procedure, you'll have lost the majority of people already. Proof isn't arithmetic and we shouldn't teach it in the same way. Sure develop the process on the easy method first, but then show people why it's important to not skip steps.