Not everyone thinks the same way. There are certainly lay persons out there who do not find proofs by contradiction jarring when they come across them for the first time. I remember I was one of them.

But the impression I get from my (possibly biased) sample is that most non-trained people intuitively see proof by contradiction (or any form of nonconstructive proof, really) as a way of "cheating", because it asserts something does or does not exist without actually producing a (counter)example. YMMV, of course.