I think it’s fine to say ‘...David and I’ because you CAN say ‘‘tis I”. Which is the same substitution of me for I.
The verb "to be" is an outlier because both subject and object are the same instance of the noun. Every other verb requires you indicate the case of the pronoun (so whether it is the subject or object of the verb).
So you say, in Agatha Christie style, "The killer, you are she!" rather than "The killer, you are her!"
Every action a subject does to itself requires a reflexive pronoun. So the dog washes itself, I calm myself, you pinch yourself.
You can send a letter to yourself (reflexive), but Johnny and Brenda can only send a letter to you (object pronoun).
A lot of the confusion comes because we dropped a lot of our distinction in cases in English centuries ago (because we went for rigid word order in English, which lessens the need to indicate case).
Although, interestingly, differences in case still exist in spoken form in many dialects. If you listen to a Yorkshire man very carefully, you will be able to hear cases vocalised differently with articles, so "Tha cat sat on thuh mat" (Tha for the subject article and thuh for the object article).
It's a lot easier to understand if you speak another European language where cases are vital for comprehension.