I don't agree with the need for stiff soles for cycling use (non-racing). That's based on my personal experience, others may have different experience.

My favorite shoes are Pearl Izumi X-Alp Seek V, which have a very flexible sole. Not much stiffer than a normal hiking shoe.

Wearing these flexible shoes, I:
* ride centuries, climb hills, and ride fast
* ride MTB and fat bike
* commute to work and wear them all day
* walk or run normally
* have comfortable feet

Wearing these flexible shoes, I do not:
* walk like a duck, or have to use cleat covers
* have to bring a second pair of shoes for work
* race
* have foot or knee pain
* have too-hot feet

Pearl Izumi seems to be the only major company that makes flexible shoes of decent quality (in their X-Alp and X-Road lines).

You can get flexible shoes from Bontrager or Specialized, but only in their lowest-end model that has otherwise poor materials.

Well, Shimano's shoes may be decent and flexible, but they are usually butt-ugly (my opinion) so I don't pay attention to them.

One drawback to the Pearl X-Alp shoes: they are wider than most bike shoes, so it's tough to fit a shoe cover over them. Pearl instructs you to use a size bigger than normal. Ok, but they don't make an XXXL cover to fit over my size 48's. So, I picked up a pair of Sidi GTX boots for cold/wet weather.