My understanding of Frank Sinatra’s connection to the mob was that he was feeding them a lot of cash in exchange for high profile gigs. His ability to send them a lot of money is why he was protected by the mob, not because he was really doing anything nefarious.
They had a symbiotic relationship. They’d find him high paying gigs and he would send them a decent percentage of the pay.
At least that is what I have been able to find out.
And yet they couldnt get him the gig for the Batman show...lolol. I can’t see him as The Joker in a campy TV series