From Wikipedia article on the Feit-Thompson Theorem proving a conjecture of Burnside that groups of odd order are solvable: "The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)


Feit, Hall, and Thompson (1960) extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory."



Why didn't finite group theorists consider groups where all centralizers of non-identity elements are solvable?
Groups in which the centralizer of every involution is solvable were classified by D. Gorenstein and various co-authors. Also J. G. Thompson classified finite groups such that the normalizer of every non-identity solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you asked about.


