The chicks from my first-ever broody hatch are now eight weeks old.

My experience is below, followed by a serious mistake I made that I'm sharing so others don't. Personal pros and cons first:
PROS: high hatch rate (11 out of 12 purchased eggs!), next to no work for me, no pasty butts to deal with (yay!), chicks are hardier, and the hen did the work of integrating them with the flock.
CONS: any chick that develops more slowly than the others is likely to be abandoned, and then you will have to raise it yourself or likely lose it; Mama may decide she doesn't like a particular chick and then you will have to raise it (this happened to me); outdoors/free-ranging chicks are more vulnerable to predation and exposure-related illness (e.g., flash downpour) than those raised indoors; first-time mama hens can make dangerous decisions (mine tried to get her week-old chicks to walk through a deep pond culvert and I had to intervene to prevent drowning/freezing - but she didn't try it again); you may have to force your broody off her nest every day to eat and drink if she won't care for herself voluntarily; you may find yourself having to re-home extra roosters or send them to the stewpot, which is something you generally don't have to worry about when you buy hatchery chicks (though keep in mind, those extra hatchery roos will be killed, just not by you); and, while Mama Hen taught the chicks that I was a friend, and they all run to me for treats when they see me, they aren't as puppy-doggish as the hens I raised myself (e.g., willing to be handled and eager to follow me around).
My experience:
My #1 and #2 top-of-the-order hens are both "super blue" hybrids from MPC. Don't know exactly what's in them, but broody breeds are definitely in the mix; they've both gone broody this summer - one of them twice. When I couldn't "break" Julia, the second to go broody, I decided to get her some hatching eggs. Didn't expect much, but she hatched 11 out of 12 of them! She was a great mama. I kept her enclosed in a dog crate inside the coop so she'd be "with" the other hens but they couldn't bother her, because a few of the others "caught" her broodiness and tried to "help" her brood her eggs.

I knew hatching was imminent when several of the other hens began hanging around the crate and acting weird (making strange cries at me, listening hard by the nest, etc.).
It's a big plus if your broody is at the top of the pecking order. I laughed so hard the day I went into the greenhouse, where I have a large dust box for winter bathing, and found five of the hens sitting in that box and afraid to come out because Mama Hen was in the greenhouse with the chicks. The big girls always ran away from the chicks because they knew Mama would get them if she saw them around the babies! Mama Hen gradually backed off when the chicks were about five weeks old and then finally abandoned them. She went off with her old flock and the chicks formed their own flock since they don't like to range as far from the coop as the older hens do...but they all hang out together when the older girls are near the coop, and they go in and roost together at night.
My other super blue, Jeanette, has now been broody for a week, so I got her some Croad Langshan eggs to sit on. I may try not putting her in the dog crate because she's #1 and the other hens don't bother her, plus she doesn't leave the nest at all except when I remove her from it for food, water, and a quick constitutional while I'm doing coop maintenance—which means I'm able to prevent mix-ups with other hens laying in her box while she's gone, or her going back to the wrong one. The other girls finally understand what's going on now, too, so they just lay their eggs in the nest box next to her and go about their business.
Now, for my big mistake: The first nine eggs hatched between 22 and 23 days after I'd put them under Julia. The last three had not pipped by the end of day three, so the morning of day four, I took them out of the nest and set them aside to throw away later. (I'd read that any eggs that hadn't hatched by three days after the first egg hatched were duds/spoiled, and to take them out so they wouldn't crack or "explode" and make a stinky mess.) To my dismay, I discovered when I went back later to throw the eggs away that one of the three "duds" had begun to unzip and one was fully unzipped, but the eggs had been cold and exposed for hours by then. The partially unzipped chick was dead. The fully unzipped chick seemed exhausted and was gasping for air so I carefully removed the rest of the shell from around it and put it under Julia. An hour later, I couldn't tell that one from the other chicks. (The third egg actually WAS a dud). So, bottom line, I likely lost one chick and endangered the other because I followed advice that said to discard un-pipped eggs at the end of day three. It's possible the one that died would have died anyway, but I'll never know. I do know I won't make that mistake again - I will be more patient and give any "duds" a little extra time this go-around.
Good luck and have fun with your broody hatch!