Nah, it goes like this:
Luffy is the captain. He ranks highest and he sets the overall direction for the crew. He makes the "big desicions" (we're going to go find One Piece; we're leaving Merry and getting a new ship, etc.) but does not give orders on everyday stuff. He also decides who gets to join/leave the crew. His orders are generally followed but can be overruled by consensus. His biggest role is to inspire the crew, lead them in battle, and give them someone to rally around.
Zoro is first mate. He is the most down-to-earth Strawhat, and the one most likely to say "wait, stop, this is stupid." or to point out the grim reality of a situation. He is the one who provides clarity and harsh truth, and keeps discipline when needed. This includes keeping the captain in line, too (like when he forebade Luffy from taking Usopp back unless he apologized) They need Zoro because he's the only one who ALWAYS has his shit together. Aside from Robin, but she's too kind to be as blunt as him.
Nami is Quartermaster, so she runs the day-to-day stuff and tells the crew what to do during normal sailing. Even the captain will listen to her most of the time.
Basically, who the "real leader" is depends on the situation. In times of despair or battle, it's Luffy. While sailing normally, it's Nami. When the crew needs a reality check, it's Zoro.
This is exactly how most real pirate ships functioned. Note also that the three leaders are also the three most senior members, which is also generally accurate.