Because they Owned him.
He was connected.
I believe this to be the correct answer. He was a money maker for them and he worked for guys who were connected that were involved in the recording industry.
If he hadn't been Italian and the mob didn't have their hands in on him as an act, they wouldn't have given a fuck. Thats why they didn't give a fuck about the Rolling Stones because they were not a mob connected entertainment act. They owed nothing to the mob.
The entire Rat Pack band was owned by the Mob and they were getting a huge cut from their shows in Vegas as well. Like you said, the MOB owned Frank Sinatra. If he would have told them to fuck off he would have become trunk music. No doubt about it.
Statements made by informant such as "Fischetti rather orders him (Sinatra)" and "Fischetti is Sinatra", or even "Sinatra being extremely powerful force in the entertainment field"....proves my previous statement that Sinatra was quite influential in the entertainment biz which was quite lucrative situation for the Mob during the 50s, 60s and 70s.
Second, the informant gives us the difference between giving an order and doing a favor. The file clearly states that Sinatra received orders from Joe Fischetti, which means someone (Sinatra) mightve being "employed" in one of the Outfits crews aka the Cicero/Fischetti crew. Again, theres a huge difference between a "favor" and "order". This statement can be backed up by the info in the second file in which Giancana threatened the Genoveses that if his proposal wasnt satisfied, they were going to take Sinatra with them (Chi Outfit) to any other place around Florida.
Again dont get me wrong since Im not saying that Sinatra was 100% made or even a high level associate or "entertaiment puppet", but still the previous files are giving me something to think about, meaning there might be some details in Sinatras career for which we still dont know anything about...the feds obviously suspected something and no wonder why theres a whole FBI file regarding his activities...