You are absolutely correct that the difference between y against x and x against y fitting perfectly demonstrates why the bias exists, but the correct way to remove the bias is not normalization, but to use a coordinate-independent regression technique.
See the other comments by many other commenters for details.
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If you plot the regression line of y against x, and also x against y, you would get two different lines.
I found it in the middle of teaching a stats class, and feel embarrassed.
I guess normalising is one way to remove the bias.
You are absolutely correct that the difference between y against x and x against y fitting perfectly demonstrates why the bias exists, but the correct way to remove the bias is not normalization, but to use a coordinate-independent regression technique.
See the other comments by many other commenters for details.