Personally, I think the reason Bairstow's place in the side is such an enduring debate is due to his annus mirabilis in 2016. He scored 1470(!) test runs that year, together with 70 dismissals as keeper - a truly extraordinary, unprecedented feat. He was one of five Wisden Cricketers of the year in 2016.
At his best then, Bairstow is probably England's best player.. but we haven't seen him at his best in Tests since then. I think we all expected him to kick on and become England's second best batsman after Root, to average over 40, and to be an established senior player.
Instead, he became a ODI/T20 machine, and won the world cup. His test results suffered. The romantic in me though thinks that Bairstow still has so much more to give this test side, and an in-form Root & Bairstow would be an incredible spine to this inexperienced batting line-up. If Buttler didn't exist, we might not even be having this conversation, because he wouldn't have had the pressure on his place.
I want 2016 Bairstow back, and I want him to smash it.