The biggest problem with a broody hen is getting a broody hen. Not every hen will ever go broody. You have no way to force one to go broody, you have no control over that. You certainly cannot control when a hen goes broody. If one does go broody and she breaks from being broody it is likely she will go broody sometime in the future, but that might be next year.
A broody hen will do the work for you, both hatching them and raising them. There are different ways to do this. I trust my broody hens and leave them alone to do their job. Many people micromanage and interfere every step of the way. Both and everything in between can be successful but that makes for a different experience for each of us.
With a broody hen hatching you have no idea what sexes you are getting. Often it is heavy on one sex versus the other. I've had hatches that were almost all male, some that were almost all female. Are you OK with that? What are your plans for the males?
Whether with a broody hen or an incubator you never know how many will hatch. It could be 100%, it could be 0%. You may read to expect 50% of shipped eggs to hatch. That is an average. My experience is more like Mathewharp mentioned. Either real good or real bad. You can maybe improve your odds by picking up eggs from somebody local and carefully transport them yourself.
With chicks from a hatchery they are alive when they are shipped, you don't have to worry about whether they hatch or not. It is possible some could die during shipping, especially if there is a delay in the mail, but the majority arrive alive.
Most major hatcheries will not sex bantam chicks but they will sex most full-sized fowl chicks. You don't get a 100% guarantee on sex, usually around 90% because of the difficulty in doing that, but your odds of getting the sex you want are much better than hatching them.
If you want them vaccinated, hatcheries can usually do that. If you hatch with a broody hen that is not practical. For most of the vaccines to take effect the chicks need to be isolated from other chickens for two or three weeks. With a broody hen they are immediately exposed. Many of us do not get the chicks vaccinated but some people do.
If you get shipped eggs they are usually all the same breed. Of course you can sometimes find an exception but you may need to look. Hatcheries can ship multiple breeds. You need to look at their pricing policies, sometimes it can get expensive to get very few of multiple breeds, but the capability is there.
If you get chicks from a hatchery you have to go through the expense and trouble of setting up and managing a brooder. Brooders don't have to be complicated but if you read too much on this forum about them people will have you going batty. You can get that with advice on a broody hen too so maybe there's not that much difference.
Another option would be to get an incubator and hatch your own. You get some advantages and disadvantages of broody hens and hatcheries.
Another option is to get POL (Point of Lay) pullets. These are usually around 16 weeks old and should be laying in about a month or so. You bypass all that raising baby chicks stuff and by then you know if they are boys or girls. They are usually kind of expensive, but so is raising baby chicks.
