Class is fluid though but race isn’t.
Race can be - both in what race someone is perceived as can differ and in what the ideas of what someone being that race means.
Class is similar - plenty of people think you can't really change class. The whole 'old money/new money' thing.
There is a perfectly adequate quantity of research papers to support the existence of white privilege.
Quantity, probably. The whole 'publish or perish' set-up has meant there is a lot of research.
That set-up, along with the current funding model and other issues, is terrible for quality research. All areas of research, but particularly the social sciences, are dealing with replication and academic journal crises. The peer reviewed process is falling apart - some of the biggest journals are having to make the biggest retractions - and in psychology, I've read something like 90% of the things we take as 'fact' have never been properly replicated and when done, the majority of 'significant findings' fail.
And even with all that research, I struggle to find anything that says teaching white kids about white privilege does anything positive. Things are more complicated than 'stating a fact, discuss = attitude and social change".
Even if you view individualistic privilege as fact, which even within CRT is still contested, would teaching white privilege that way be worth it if all it does is make white people hate other white people more? If it did nothing to tackle racism, nothing to bring people together? Because as the study I linked before suggests, there is evidence that that is all we'd get. If education/facts by themselves did much for social and behavioural change, we wouldn't have an obesity epidemic.
There is also a glut of research that discusses racism and other forms of oppression through lenses other than individualistic critical race theory. We can discuss social issues at a population and community level, we can discuss history, systems, and diversification of the methods of change. We can do all of that without ignoring that even within some branches of CRT, privilege isn't this fixed thing people have -- if social systems change, then the social (dis)incentives, social openings and barriers, those all change too.
I know Western models of societies love to get individualistic, that our education and media love their born-underdog heroes and born horrible villains. There may be ways to use that to our advantage, but really, to get people involved in structural, systemic change, we have to discuss issues as structural and systemic, we have discuss these as actions perpetuated by systems and that we can change the systems, not something some has or doesn't have.
And, as noblegiraffe said, there are a lot of people on this site so lots of people discussing one thing can just mean those with other opinions don't get involved. It has theoretical merit, but it's practical applications haven't been shown. I've seen individualistic male privilege being taught to kids backfire spectacularly in my own work much like in the white privilege study I referenced earlier. It didn't make anyone nicer, it didn't bring anyone together, it just fed more into this individualistic way of looking at the problem and started to treat each other and themselves as the enemy which I don't think helps anyone.