As such

Where I live, it is common to use 'As such' to mean 'Therefore'. But according to what I have read in an English usage book, this is not correct.

He is fat. As such he cannot run fast. (This is one example of the phrase used wrongly.)

The phrase is correctly used if 'As such' refers back to the subject.

She is the principal of the school. As such, she has the authority to cane students who have committed serious offences in the school. ('As such' refers back to the principal.)

I wonder whether native speakers are in agreement with this rule.

Many thanks.