The acceptability and the style of the syllepsis in 'He threw stones and then pu...

I wonder how native speakers would perceive the style of this sentence and how they would react emotionally to it:

He threw stones and then punches at the protesters.

How acceptable and idiomatic is this sentence?

According to sources, the OP's sentence is indeed an example of syllepsis. This is a rhetorical device that writers use to amuse and surprise their readers. The same word is used in two different situations, and more importantly, the end result is always grammatical. Examples taken from Thought & Co

1. When I address Fred I never have to raise either my voice or my hopes. _{E.B. White}
2. Piano, n. A parlor utensil for subduing the impenitent visitor. It is operated by depressing the keys of the machine and the spirits of the audience. _{Ambrose Bierce}
3. It's a small apartment. I've barely enough room to lay my hat and a few friends. _{Dorothy Parker}

However, the OP's sentence should be rewritten for the sake of clarity

He threw stones and then punches at the protesters.

In the sentence above, the adverb ‘then’ spoils the attempt at syllepsis, it splits the statement into a sequence of actions, making the word punches appear to be a verb.

By removing "then" the verb "threw" can apply to both objects at the exact same time. The reader immediately understands the literal meaning (throwing stones) and the metaphorical meaning (throwing punches).

He threw stones and punches at the protesters.

The preposition at can be used with “throw a punch” if the target of the aggression is a person or thing. Whether modern readers would be amused by the statement is debatable, but it would be understood.