The possibility of using wāw when the pronoun is absent does not mean any complete sentence can be made a circumstance of any noun. The content must plausibly describe a state attending the event in the host clause.
In جَاءَ زَيْدٌ وَالْمَطَرُ يَنْزِلُ, falling rain can provide the temporal circumstances of Zayd's arrival. Two wholly unrelated facts do not automatically become a genuine circumstantial relation merely because wāw has been inserted. Syntax, time and context must support the reading.