I In speaking you diagrammed sentences till you understood the relationship of sentence structure and you don't use them again. You say bob and I went to the store. because broken down it is bob went to the store and I went to the store,,,, the and combines two thoughts into one joint thought. and you don't say "me a a bob went to the store". because..... bob went to the store is good but me went to the store is just plain use of words. Yet how many talk like that ?
I think your analogy is apt, but not for the reason you think. The reason you give for "Me and Bob went to the store" being incorrect has nothing to do with learning to diagram sentences. Research has indicated that being able to diagram sentences makes a person able to diagram sentences, but not much more than that. What it teaches is soon forgotten and rarely used, as it was in this case.
With sentence diagramming (or other methods of teaching those formal rules of grammar), you would have come to your conclusion differently from the way you describe. You would have recognized that the combination of "me and Bob" is a phrase functioning as the subject of a clause. It could have been the main clause of compound sentence or a compound complex sentence, or it could have been a sentence in itself. You recognize its function is the subject of the clause by the fact that it is performing the action of the verb "went," with the remaining words forming an adverbial prepositional phrase modifying the verb and answering the question ""where." You would have recognized that while "Bob" is a proper noun, "me" is a pronoun, and pronouns must be in the proper case. Since the pronoun is part of a phrase functioning as the subject of a clause, then it needs to be in the nominative case rather than the objective case. Since "I" is the nominative case and "we" is the objective case, "Me and Bob" is incorrect.
You had instruction that taught you that, but it is not what you remember, and it is not the method you described using. You also had a teacher who taught you a very simple shortcut in lieu of that system: when you have a combination like Me and Bob, if you take them individually and listen to the result, you can tell if one is wrong. How can you tell? Well, by the fact that "Me went to the store" sounds wrong. You know this because your brain internalized that fact over years of listening to the way people speak. Linguists believe that children internalize 90% of the rules of grammar by age 5, long before they have had any formal instruction, although they could not put those rules into words. In fact, people learned to speak, read, and write English before the rules of grammar were invented. Shakespeare never had a single lesson in formal English grammar, because it had not been invented yet.
So that is one of the reasons that sentence diagramming is almost never taught these days. It is hard, in fact, to find a textbook that includes it. (I used to purchase such textbooks.) It teaches very little that you really need, and almost everyone forgets it.People need to use need some other way to decide what is correct.
I took my very first dive trip to Cozumel back in the last millenium, and I pulled out my newly-learned tables to track my dives. Everyone who saw me burst out laughing. The multi-level dives we were doing made the tables useless there. If you wanted to dive in Cozumel, you had to follow a DM by law, and even back then those DMs were using computers to track their multi-level dives. I had three choices. 1) I could just follow the DM and trust him to keep me safe. 2) I could hire a private DM to lead me on square profile dives. 3) I could get a computer.
That time that I pulled out my tables in Cozumel was the last time I ever saw any recreational diver use recreational dive tables anywhere in the world. They work for square profile dives, but like sentence diagramming, most people have forgotten how it is done and have chosen a different system for tracking and planning their dives.
