I am searching through my sources. In one, I found this explanation
Do you understand what it means to cite a source? It means to state the name of the author and the title of the work. You haven't done either of those things.
which confirms your claim:
The explanation for what you wish to reduce to "my claim" is that sentences like "I [[went to the beach] and [learned to swim]]" and "John [[cleaned the department] and [cooked the dinner]] contain coordinated verb phrases, NOT coordinated clauses, let alone coordinated independent clauses.
There is no question of an "implicit subject" of the second verb phrase. It isn't implicit. The subject of the second verb phrase is the subject of the first. Both verb phrases have the same subject. There is one independent clause with two conjoined verb phrases. That's all there is to say.