Some of the claims are that they control philosophy, science, technology, politics, and entertainment.
Aren’t these claims that require thorough investigation?

No. If I told you there was a gang of demons controlling the army and the UN, would you go out and conduct a thorough investigation? This is Russell’s celestial teapot example.
Beyond that, I think our attitude toward those spheres (media, technology, politics, entertainment) should be based on their own substance. If I disagree with what is happening there, I will criticize it, and if I agree, then fine. The question of who is behind them and who is driving them is not really important for the purpose of how to relate to them. It might matter in order to know how to fight them if you have decided that this is what should be done, and here we return to my first paragraph.

With God’s help, 23 Tevet 5780

To Raphael — greetings,

“Builders” are Torah scholars, who are called “builders,” as it is written: “Torah scholars increase peace in the world, as it is said: And all your children shall be taught of the Lord, and great shall be the peace of your children; do not read ‘your children’ but ‘your builders.'” Torah scholars are not only “builders” but also “free,” as it says: “You have no free person except one who engages in Torah.”

And indeed Torah scholars aspire to impart the Torah to all who enter the world, and therefore they ask in their prayer, “And all the living shall thank You forever.” And at the conclusion of their prayer they ask: “May all the inhabitants of the world recognize and know that to You every knee shall bend, every tongue shall swear; before You, Lord, they shall kneel and fall, and to the honor of Your name they shall give preciousness, and all of them shall accept upon themselves the yoke of Your kingship.” Then the entire world will merit being built through freedom.

With blessing, Perry Mason