In «ضَرَبَ زَيْدٌ», Zayd is an overt subject. But «ضَرَبَ» can mean “he struck,” with an implicit subject pronoun «هُوَ» recoverable in context.
When the overt subject of a tanāzuʿ construction is assigned to the second verb, the first verb’s subject must be accounted for. An implicit pronoun is a syntactic analysis, not an extra written word.