Tossing in my two cents.
My coop is maxed out with chicken numbers and just doesn't need more chicks running around. So while it would be nice to let my Miss Broody Butts go au naturale, it just isn't practical. When hens go broody, something in their brains just snaps. They aren't normal chickens anymore, they are nuts.
Letting a hen go broody and NOT sit on eggs, can lead to death. Sitting on eggs to hatching lasts 21 days, and while they do eat and drink during that time, the amount of weight they loose is shocking. When they are NOT sitting on eggs, but acting the part, they'll sit long past the 21 days, and can even starve to death.
The broody breaker cage hasn't worked here, but I would always provide food and water at all times. The cage has small hardware wire for it's bottom to allow cooler air to circulate around their warm bum.
The idea being, that the cooler air constantly cooling them off, helps turn off their nutty switch.
I'm not a fan of the ice eggs substitutions, the ice in a bag in the nest box, or the frozen bottles of water method. I think it would be too cold, but my biggest problem is the moisture. The added moisture from condensation or melting gets into the nest boxes, and makes a mess with the other birds. Dusty birds, and wet = extra dirty eggs. (my broodies switch boxes, and other hens have no problem sitting on top of a broody, when you gotta go, you gotta go!)
Removing them every time they crawl into the nest box works for one of my hens, but you have to be diligent and do it all day long for a couple days.
What has worked for my stubborn Lavender Orpington and my Olive Egger is a cool, NOT COLD, bath. We have 100' days AND my hens are used to show baths. So going into water isn't traumatic for them. They actually just sit in three inches of water, feet resting on a washcloth, sitting in a large old roasting pan, like a duck (not floating, sitting). I sit with them the whole time. There is nothing in the water, but water. After 5-10 minutes, I lift them out, let them drip for a second, and spend the afternoon drying and preening. Two to three days, and no more Miss Broody Butt.
Eggs laying comes later. Broody here isn't about egg drop in production. A broody sits for three weeks and kicks back into laying about 3 -4 weeks later, so 7 weeks of no eggs from that hen. Stopping a broody, and you can still be without eggs from that hen for 3-5 weeks. HOWEVER, if you have more than one hen going broody, OR you are maxed out in your population or breeding program, all those non-layers can really hit the pocketbook hard at the peak of the season.
So I guess it boils down to finding something that works for you, works for the hen you are dealing with it, and being patient with the process.