Only one tweet of Vallelongo's has been controversial. Unless you can produce the "couple more" you've seen, we must assume you're misremembering. Literally nobody involved in the controversy has claimed that the "entire story" of Tony Lip is a fabrication. As for the claims of Dr. Shirley's family, the man himself is on tape saying the following:
"I trusted [Vallelonga] implicitly. See, Tony got to be, not only was he my driver, we never had an employer-employee relationship. We didn’t have time for that. My life was in this man’s hands. Do you understand me? So we got to be friendly with one another." (https://www.vanityfair.com/hollywood/2019/01/green-book-director-peter-farrelly-defends-movie-against-criticism)
Evidently the family didn't have a clear understanding of the relationship between the two men.
Finally, Dr. Shirley did not "fire him almost immediately." In fact Lip's employment with Dr. Shirley lasted 18 months—much longer than the two months shown in the film.
