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The 3 schools against skepticism all argue against each other's points, so in a sense they strengthen the skeptical position.

Appears there is no clear consensus on exactly why skepticism is false.

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Your argument is an argument of Pyrrho, 3rd century BC. It refutes itself if you think about it. What they have in common though is making or relying on strong arguments against skepticism. The question is which alternative is the best, not whether an alternative is necessary.

Can’t one turn this argument around to any other school of philosophy?

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