This is really just based off rumor and speculation but I think the issue may be not just Southgate picking his favorites but essentially not appreciating White as a defender. White has played most of his career as a CB, he really should be in contention for those spots in theory along with RB. Yet in the 2021 Euros he was behind not just the starting group but also Tyrone Mings and the rumor is that in the 2022 World Cup he was essentially informed from the start that Eric Dier and Conor Coady were preferred ahead of him. White was one of only three field players not to see a single minute on the field during the Euros and the World Cup was shaping up to be the same story.
I have time for the argument that a player should just suck it up no matter what but, yeah, the manager is going to piss a player like Ben White off if he is burying him not just behind Harry Maguire but also the likes of Mings, Coady, and Dier with seemingly no hope to change that situation. That's not exactly Rio, Terry, and Sol Campbell.