Well, I'd always write "People who think I'm a pedant are wrong" because there I'm thinking of people as individuals, as humans, who happen to be several in number. "People that think..." is patently wrong because things don't think, unless they are computers that 'think'.
However, I might well write "people that" when they are not being considered primarily as sentient humans whose humanity, emotions or thoughts are the subject. of my thinking e.g. coldly, as the subject of some article in economics where they are no more than a sector or group 'that' is subject to analysis.
That said, 'people that' has history. The hymn "All people that on Earth do dwell" was written in 1571 and has been in English hymnals ever since ! .