I have not done the Bailey Range, but did a conventional Olympus-only climb 3 weeks ago (Sept 18ish). So I can speak a little to the wetness/snow + glacier aspect, but not to the rest of the route and its sidehilling, cliffs, etc. In general, my trip was easier on the feet/ankles than yours would be.
We spent the first few hours in the dark on Blue Glacier in wind and driving rain. According to the last forecast I'd heard, snow line was supposed to be hovering close enough to descend to our level, but it did not. Around sunrise, rain moved out and it was just windy the rest of the day. The three guys with me wore their hefty waterproof mountaineering boots.
Call me crazy (they did :), but I was wearing Salomon SpeedCross 2 shoes (no gore-tex) with crampons and old non-waterproof busted-up Threshold gaiters, and light Smartwool PhD or similar socks (ankle height). My feet obviously got wet, but never cold, although I did make a preemptive switch to a heavier pair of PhD socks (medium? weight hiking socks) + some high-tech VBL (plastic bags, but outside socks) a couple hours in because I was a little worried they might go towards cold territory, as the weather was causing a lot of standing around while trying to navigate. Moving feet are warm feet in my experience, but standing feet can cool. They stayed plenty warm and the new pair of socks/liners worked fine and later I contemplated switching back, but didn't have a long enough break at the right time. After the rain stopped and the sun came up, my shoes were mostly dry (or not noticeably wet) for most of the rest of the day. Back at camp, I had not even a hot spot. The boot-wearers had a combined litany of painful-looking blisters and abrasions. 1 or 2 of them had also suffered cold feet (the literal, not the figurative, variety) during the climb. (I suspect due to tight boots cutting down some circulation?) Take this as a report of what happened to our feet, not a sermon.
I weighed this choice of footwear very carefully beforehand, consider expected conditions, unexpected conditions, and previous experience. It worked for me, but I would not have recommended it for my companions, for example.
Now, if I was carrying a heavier pack, I likely would have gone for boots for the support with crampons and potentially hard snow, crampons on rocks (yuck, hard on the ankles), etc., but really, my feet appreciate shoes over boots as long as they're carrying weight within a reasonable window of my own.
Also, from my experience, I suspect I have more resilient ankles than others.
And I'm a newbie to the mountaineering end of things, reluctant to let my light backpacking style slide. Don't listen to me. :) Listen to these other guys with way more experience.
But for the approach (obviously much tamer than the Bailey Traverse), shoes hands down.
On a previous trip that included 2 days straight of snow/rain/wind on tamer terrain, the wet feet in mesh trail runners were annoying, but not cold. And the SpeedCross setup also worked on Baker, but the conditions were tamer there.