I've tried a few times to get a hen to go broody by increasing the eggs in the nest one a day until I get to a dozen or more, then just leave them. Sometimes I mark real eggs and sometimes I use fake eggs, golf balls. One time I did get a hen to go broody, but that was on a different nest, I don't think that counts. Those hens were hatchery chickens or chickens I had hatched from hatchery chickens. A few had gone broody in the past, my Black Australorps. My Buff Orpingtons never did go broody. Some Ameraucana I got from a breeder went broody a fair amount too.
I purposely hatched eggs from the hens that went broody and kept some as replacements. By keeping my replacements, rooster as well as hens, from the hens that went broody I now have a flock that most of the hens go broody, some of them a lot. But I cannot control when a hen will go broody. To hatch eggs when I want I use an incubator.
I don't like just leaving real eggs in the coop overnight although I did it a couple of times to make sure it wasn't just fake eggs that did not work. Eggs left in the coop overnight are an invitation to predators, rats and snakes if nothing bigger can get in. I don't think they store real well for hatching anyway, though if the nest is on the ground it might work better. The ground will help regulate temperature. Instead of storing up eggs hoping a hen will go broody and maybe keeping them way too long, I wait until a hen goes broody and then start saving eggs for her to hatch.
That creates a problem if you want to hatch eggs from broody hens though, so maybe collect the eggs you want her to hatch and store them in an egg carton in a relatively cool place. Every day take out the oldest egg and save the newest. That way you always have fresh eggs to set. But as others have said, you may go months waiting for one to go broody, if one ever does. That's why I like my incubator, I control when I hatch.