>but I cannot find a single flaw in his arguments.
The defeating argument against "Wildbergerist mathematics" is that it's decidedly informal. He doesn't provide formalisations and upon being pressed refused to be more formal about it than he is. Which is to say, he doesn't want to explain his system in formal logical terms as it became a tool after around 1860 in mathematics. One doesn't have to accept first- order logic, a variant of it like second order, or something more exotic that exists. But the guy doesn't provide any formal system encompassing his rules of inference and he knows it. The problem with this is that whatever is allowed or not in his system remains up to him only. He champions a form of constructivism, and rejection of infinities, but it's not like he affirms strict Heyting arithmetic either. He didn't engage with intuitionism proper (which conflicts classical logic also), but he also didn't provide a formalized alternative. He doesn't provide a jumping off point for other people to expand on Wildbergerism.
I like him, he has good videos, good ideas even, he has some breadth. But by his own design, his system can't be taken over, since what mathematical rules he likes - some subset of standard math - is spread over his videos and he has no formal system as we'd know if from the mid 19th century onwards. What remains is inspiring videos where he breaks down criticisms of classical mathematical system which were already fleshed out by Kleene, Heyting and the likes - except those guys worked out fully formal alternatives.