Well remember that soldiers or even associates can have their own crews.
Larry Gallo and Joey Gallo were both made members of what was then the Profaci Family along with supporters such as Carmine Persico and John "Mooney" Cutrone. Larry was considered to be the crew leader. But at the time of the troubles, these four men (and who ever else was "made" ) were officially soldiers, not captains. I've read that they officially reported up to Frankie Abbartemarco, but also had understandings and up until then good relations with captains Johnny Oddo and Harry Fontana.

The Gallo crew was very large and was allegedly distinctive for the time for having a large number of non-Italian associates, who of course would not have eligible for formal induction.
I think the initial goal of the Gallos was just to get more money and/or keep more of what they earned instead of having to pass it up in tribute. Joey may have had more revolutionary ideas but ultimately it was Larry who ran things.