Having bankai takes anywhere between infinity and two days, MASTERING bankai takes somewhere in the vicinity of ten years, or longer. There are captains who have yet to master their bankai, including the all-time-favorite known for Taking It Alone, so I can see no reason why Ikkaku couldn't be captain, except he won't allow for it. Renji, by the same logic, could also become a captain but his behavior and ties could potentially betray his loyalties to Soul Society. Rukia has barily met the stage where she's mastering her shikai, not a fucking prayer in the universe for her, and also she has history that's bad for captainship. In fact, Ol' Yamma might have nixed two or three More captains if he hadn't already lost three in a single day, just for concerns of dependability.
A rock-awsome captain is pretty damn hard to come by. Even the shittier captains in SS are not only powerful, but skilled, determined and Deeply loyal. Even batshit-insaneo-boy Kenpachi respects and obeys Yammamoto.
Also, given the current openings who out of the listed potentials would really be appropriate? We need someone to take absolute command of the Psychological Warfare division, the Peacekeeping division, and the Human Relations Division (if I got the numbers correct). If nothing else I don't think Psych is an easy fill.