The benefits are enough that data structure and algorithm designs in the JVM compiler world often take advantage of assuming reducible control flow even though Java bytecode can express irreducible CFGs. Instead, such programs are left to the interpreter.
Most problems on reducible graphs are not easier than for general graphs, because you can always compute a loop forest (for generalized loops, not natural loops with a single entry point) and just consider a derived acyclic graph.
> you can always compute a loop forest
It's easier to not have to.
https://pp.info.uni-karlsruhe.de/uploads/publikationen/braun...
In the case of irreducible control-flow it may produce redundant cyclic phis, but those can be eliminated with a pretty simple post-pass.