The tamest chicks I've ever had were hatched and raised by a broody. I have no idea why they are tame--I really didn't do anything to tame them other than I gave them live meal worms a few times.
I don't believe you need to remove chicks from the broody to get tame chicks. I have found much of a birds personality is genetic and can be 'tweaked' with human interventions, but a flighty bird is a flighty bird and a calm or curious one is going to basically remain that way. Having many, many broodies with chicks to base my findings on....
What I do find is needed to improve people/chicken relations with a broody is dedicating time to it and finding a treat that they enjoy. I don't mean hours, but it does take short blocks of time a couple of times a day.
My choice of treat is a mix I make myself of a quality wild bird seed (I think it is 18 or 19% protein) with a mix of berries, nuts and seeds available at Lowes or
WalMart. To a couple of cups of that seed I add a cup of finch seed, a cup of dried meal worms, a half cup of hulled sunflower seeds (available as a bird seed at
Walmart also) and a cup of my normal scratch which is from a feed mill and has crushed corn, wheat and some oats. I mix my 'Sweet Treat' up in a large batch and then just fill a sandwich bag with a couple of handfuls and tuck it in a pocket before going out to the coop.
A broody has a couple of overpowering instincts... protect her chicks and find them food.... even the flightiest hen can eventually come around if you keep her in the run with the little ones, go inside and sit on the ground and scatter a bit of treat close by. If she isn't willing to come close you may need to scatter some close to where she is at then move a few feet away to sit, it won't take long before she begins to realize you have something she wants for her chicks. I have found most chicks tend to be curious and will sometimes be willing to approach you before the hen even is. Don't try to reach for the chicks or otherwise bother them, let them come to you and enjoy their treat undisturbed. Eventually you will be able to keep some treat on your hand and lay it flat on the ground and let the chicks eat from your hand, then scatter some on your leg as you sit to see which ones are brave enough to jump onto your lap to get a goodie. If you do this for a few days it will teach the chicks that humans are the source of something good. If you try to grab the little ones or hens during this process then you will loose all of the progress you made and maybe more, so patience is needed.
Another point... always use the same sound to call them while teaching them this way... a soft 'pook, pook, pook' or 'here chicky, chicky, chicky'.... something simple and rhythmic that they will begin associating with a goodie being offered will help immensely when they are older and you want to call them in, but when you call them make sure you have at least a handful of something good to scatter for them. Birds are food motivated. period. If they think it will involve food they will respond if they were trained with repetitive exposure.
Once the chicks willingly approach you on the ground then begin to call them as you approach and scatter the treat near you when standing and allow them to come the rest of the way to you. They will soon be headed your way when you walk into view just to see if you have any goodies to share.
After the chicks are willing to come around you can work on making them friendlier. They should be gently handled and offered a goodie when picked up so they view it as a positive and not a negative, keep it short and simple to get them used to it, but again, time is the key and not all chicks are destined to become 'lap chickens'.... it just isn't a common personality trait. But you will have birds who are used to being handled, will come when you call and don't freak and run the other direction when they see a human.