Does "גבר חזק" have any association with dictators like "strongman" does in English?
Why don't you need "את" the direct object indicator in this sentence?
Would it be wrong to use it?
You only use את to indicate a definite direct object of a verb. In this sentence, there's a direct object of a preposition.
The basic meaning is indeed "on": אֲנִי כּוֹתֵב עַל נְיָר (I write on paper). But if you are not a tatooist, you propably won't write on strong men. The content you speak, think, write about is expressed in Hebrew with עַל too: אֲנִי מְדַבֵּר עַל הַסּוֹד (I speak about the secret). Think about it as "to write on (the topic of) ..."