Control might be a bad term, more like they had the most reliable, consistent, dependable pipeline. At all points, they had big influence, from political connects in Sicily, to mafia family members with access to opium growers in turkey, mafia family members in the U.S. To receive the dope, connects to the teamsters to move it around the country, and a ready made distribution network of costa nostra crews to move it. They also seemed to have been open to working with African American gangs to further facilitate distribution.
You make money in drugs three ways, distribution (ormento, Nickey Barnes)
Importation/transportation (think MIckey Munday, maybe Carlos Lehder, the Mexicans after Medellin, original Bugs n Meyer gang, though that was liquor)
And production (French Corsicans, Colombians,ect...)
At the time we are discussing, THe five families seem to possess maybe one or two of these components, but DEtroit is seemingly the only one with all three firmly in their control. It seems getting opium from the Middle East was a kind of cutting edge thing, as was previously stated, for years the stuff was coming from China and whatnot, through Jewish gangsters.
I took the three components I listed, and by process of elimination, tried to s e who in New York had all three, to me Joe Adonis looks good for it, but EVEN HE has Detroit ties with ford so....