There is a particular and subtle twist going on here, that probably helps illustrate your point. Just because the shooter identified as white (if he did) doesn't mean that most white supremacists would agree with his self identity.

White supremacy and racism are gates maintained by strict gatekeepers. The shooter was, by accounts I've read, of Mexican American heritage. White supremacists and racists in Texas focus their racism on Mexican Americans. So, whether the shooter thought of himself as white or not, we can be sure that lots of racists in Texas held the opposite opinion about him.

Race is complicated in the United States. Cuban Americans, for example, who are most densely represented in Florida, are more likely to be accepted as white if they "look" white.

I believe the reason is myths that circulate about the noble European heritage of formerly elite (wealthy) Cuban families who fled Castro's revolution.

In the minds of racists, Cuban immigrants who look white are good white people who fled persecution and economic chaos created by people of color.

The reality is far more complex than such a simple, ahistorical narrative, but historicity has never been critical for white supremacists.

By comparison, we see a narrative in which even Mexican people who "look" white are usually not accepted as white by white supremacists. The ahistorical narrative going on there is that Spanish colonizers mixed "too" freely with native Americans, thus "corrupting" their bloodlines. That sort of talk is common in racist corners of the Internet, and when I was younger, I used to hear it in real life.

Such open talk is less socially acceptable now, but reactionary forces are certainly working to bring it back into respectability.

But anyway, that's why it's perfectly possible for some people to consider themselves white while being rejected by gatekeeping racists.