A complicated message for a group of people who were rounded up on racial grounds and then murdered en masse. The problem here is that when something doesn't fit a skin colour narrative it becomes about 'whiteness' - and the whole argument becomes redundant because that word basically means whatever someone wants it to mean to further their argument.
And Jewish people do control the media... in Israel. White Supremacy doesn't control the media in China, Japan, Africa, South America etc. Arguably nobody controls the internet. It makes sense in the US (where I guess that you're writing from) that WASPs appear all over the place in the media but they are also the most populous group.... that's like saying Bollywood is dominated by Indian people. These things are demonstrably true but that doesn't make the point necessarily relevant or helpful.
And it's not quite as bleak as you suggest. The judiciary for example ends at the Supreme Court. It's current iteration.... four white guys, a white woman, a (white presenting) Jewish woman, a black man, a black woman and hispanic woman who identifies as Nuyorican. Not entirely a WASP fest - but room for improvement.
If we're going to have loaded discussions about race (and I'd prefer we didn't, because I think identity politics is a zero-sum-game pissing contest) then the Jewish question is a nuanced, complicated and important one. If you get to decide what constitutes 'a good race' and what constitutes 'a bad race' then you need to let Jewish people self-identify on this one. Do they consider themselves white? The answer seems to be is that it depends. The fact you consider them white is (according to the batshit rules of identity politics) an act of racial aggression in and of itself. In much the same way that someone decided they weren't white in the 1930s then made some fairly assertive policy decisions off the back of that.