But quantum gravity works very well. Did you mean: "quantum gravity which should work in my opinion doesn't work"?
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Why 10 dimensions? Because that's a direct consequence coming from the basic postulates of relativity and QM. So you may ask God. There's nothing strange in extra dimensions in the same way as there is nothing strange in extra fields in some QFT.
That’s simply not true. 10D is not a direct consequence of QM. If you can show it follows directly from the axioms you’ll get a Nobel prize. I invite you
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Extra dimensions turn out to be necessary when we quantize the string and impose some constraints resulting from QM. That's very elementary derivation which you can find in any string theory textbook. So yes, extra dimensions are definitely quantum effect.
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Yes they are necessary to formulate string theory, however what’s not necessary is the radius of the extra dimensions. The reason they “must” be so small is because the others have been ruled out
End of conversation
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