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gophersnake
The time I took those Mist Trail pictures I was wearing Tecnica Hermano cross-trainers. I've since switched to Merrell Moab Ventilators. Either shoe gives me great traction under all normal conditions but nothing works under all conditions. I personally find sandy rocks more of a challenge than wet ones (but I'm also on sandy rocks more often than on wet ones). I'd think that wet rocks with sand or mud on them would feel different from clean wet rocks, and wet rocks with algae would feel different still.
Conditions on the Mist Trail are going to vary from place to place and from season to season. If you wore something that was ideal for clean wet rocks, you might have an even harder time than with ordinary shoes on sandy, muddy, or algae-y ones.
My favorite traction aid is trekking poles with carbide tips. The way I use poles on slippery stretches, I'm basically walking four-footed. When I have two poles firmly planted downhill from me, my feet are still likely to slip occasionally but I can take my time choosing the best places to step, and it's no longer critical if I do "skritch" occasionally.
The biggest problem I can see with carrying a kid is that you wouldn't have both hands available for poles. Even one pole (staff) would be better than nothing, though.
That's close to what I've used when I've been on the Mist Trail. The first time it was some Merrell backpacking boots with Vibram soles, and only going down. I did like the poles to steady myself. The second time was round trip wearing circa 2006 Nike Air Zoom Tallac + hiking boots. These were odd looking (looked like you could play basketball in them) but did have a grippy low-wearing sole. They were actually like climbing rubber. On something like a tile surface they would make a sticking sound and I'd find lots of stuff stuck to them. My wife wore Merrell Siren shoes (she really likes them and doesn't like waterproof shoes).
As for my kid, we've still got a pair of kids' Merrell Moab Ventilators that should still fit by June. That's if the little guy feels like walking. They're not like the adult version. They use laces that can't be removed and that cinch with a toggle. They were cheap too - in the clearance section of a Stride Rite outlet store (Wolverine owns Stride Rite and Merrell).

Oh - I just got in the Five Ten Guide Tennies today. They fit pretty well, especially considering I've got wide feet. I noticed the laces extend really long - I guess like how climbers like it. The toe area seemed a bit hard at first, I'm wearing them around the house to try and break them in. I tried to see how grippy they are (and they certainly are). There's stuff from the kitchen floor stuck to the rubber, and when I just used the refrigerator door in place of a climbing surface, it left a black mark (came off with rubbing alcohol). The rubber also has a funky smell like brand new tires. It's also an older model with only the dot surface. I understand the newer models have a specific molded surface with a more solid front and "Stealth" molded into the rear.

I understand this is probably overkill for just the Mist Trail, but I'm not a climber and don't plan on doing a lot of scrambling. I always wanted a pair of these because I thought they looked cool.