The call that decided it
A hire I made a few years ago was decided on a phone call with someone she used to work with, not in any of the interviews.
By the final round, both candidates were strong. Their resumes were polished, their answers were polished, and I couldn't tell you which one would be better on a hard week. So I called someone each of them had worked with. One reference said nice, general things. The other told me about a launch that went sideways and exactly what my candidate did the next morning. That was the hire.
I think about that call more now, because AI has made everything before it look the same. Resumes, cover letters and even interview answers can be prepared to a high polish by anyone. When everything you send looks great, hiring managers go looking for the one thing that wasn't written for them, and that's what someone who worked next to you says when you're not in the room.
Most people treat references as paperwork at the end. I'd treat them as the most important part of the search to prepare for. Before you start applying, pick two people who saw you handle something hard. Call them, remind them of the specific moment, and ask if they'd tell that story if someone calls.