My experience with the bandannas is that they actually make it worse. Unlike the evaporative vests, which are very thin - a layer of water-repellent parachute-type fabric inside, a very thin layer of absorbent fiber between, and an outer permeable layer - the bandannas are just full of those beads like you put flowers in. So the outside stays cool, but your neck is actually insulated and hotter than it would be bare.

I use an evaporative vest on the moto and I don't doubt it's saved my life more than once. Before I got it, there were a couple of times when I overheated, got dizzy and indecisive - not good on two wheels in traffic. But I think that's more related to the protective clothing motorcyclists and mountain bikers have to wear (like the picture you linked to). Never had an issue on the road bike, with the wind I generate. I'm a lot more likely to get in trouble running, where there's just really not a lot you can do but be careful, slow down, and be willing to bag an interval session.

Only people I've ever known with the active cooling vests, besides the professional auto racers you read about, are people with MS who need to stay cool for health reasons. Way too much weight for championship motorcycle racing, though I wonder whether endurance racers and rallyists might wear them.