Gilgulim is interesting because some Rishonim and Acharonim have declared it to be a very important yesod and some have said that it is Avoda Zara. I wonder if it is possible that there is a middle ground (i.e. gilgulim do occur but aren’t that important).
Interestingly reports of gilgulim have been studied pretty extensively and there is substantial evidence that they do occur. A Professor at the University of Virginia, a psychiatrist, extensively researched numerous cases, and published many articles and books on the subject. A colleague of Stevenson published a book called “Life before Life: Children’s Memories of Previous Lives” which summarized the results of their research at the University.