Actually thats how I do it, and I am glad to hear others do the same, with a couple of exceptions....
I do a split WITH the old queen, but then I put a queen IN the old / production hive. As mentioned they feel they have swarmed, and so long as population is managed they will not be likely to actually swarm.
Getting older bees to accept a new queen is not as easy as getting younger nurse bees to accept a queen, so a little more care and caution is in order when putting the new queen into the production hive.
This is why most people, and publications talk of adding the new queen to the split rather than the old hive.
I like the idea of letting them raise their own queen, but rather than letting them raise their own queen, why not notch a cell or two on two different frames (if you use that smelly plastic foundation) and convince them to raise queens for however many hives you have, OR, do cell punch (If you use real wax) and drop the bars into this hive and let them make queens that way? Then you get new queens for how ever many hives you have. Put the old queen in a nuc, drop the cells into the hives and wait to see if the new queens mate and begin laying. If they get eaten on their mating flight you can reunite the original queen.
Simplest way to keep bees ever.
We have farmers who... HAVE... bees, that do NOT manage them beyond tossing a super on in the spring and pulling it off in fall. When they die they make some calls and toss more bees in there. Sometimes they get honey, sometimes they don't.
Living in an urban environment puts more responsibility on the beekeeper because they have to be concerned about neighbors. SO, what to do to prevent one of your swarms from hanging off the neighbors front porch?
Keep a young queen in your hive. Requeen EVERY year. Younger queens are better at preventing swarming tendencies.
Remove brood frames. When your queen has five or six frames filled with brood pull a couple of them, replace with either drawn comb or an empty frame. This will help keep the brood nest open so the queen has plenty of room to lay, and it will reduce the number of bees emerging.
I have pulled 4 frames from hives that have had 7 frames packed solid with brood. Slide the brood nest together and place the new frames to the outer edges of the brood nest.
What do do with the brood you just took out?
You have a couple of choices. Make more hives, strengthen other hives, or destroy the brood by Decapping or freezing.
DONT, use your hot knife to decap the brood frame. If you do you will spend a lot of time with a brillo pad cleaning it.
IF NO other hives/nuc's are needed, or wanted. Use a filet knife and shave the capping off those frames. Cut them down well, and put them back in the hive. The bees will immediately go to work cleaning that mess up. As they clean it up and polish the cells for re use, the queen will lay in them.
Freezing works too, but in my opinion is more work for the bees. they have to open all the cells to drag the remains away, plus it means putting that frame in the freezer over night. Much easier to decap and drop the frame back in.
I am not sure why it works so well to do this, perhaps, because it suddenly gives the bees SO MUCH to do that they don't feel useless anymore?
It will have to be done twice, MAYBE three times during the prime months depending on how prolific the queen is.
Thankfully I want more bees. SO making the nucs is of prime importance. With ten Nuc;'s it is really easy to rapidly build up a newly started hive. Package, or hived Nuc from that spring with the over abundance of brood from those nuc's and other hives turns them into booming monster production hives, usually in time to take advantage of the spring flow.
If hive has good brood pattern and a decent number of bees, I have never had a problem with not enough bees to cover a frame of brood. If the hive is thought to be too weak to cover the brood, I give the frames a LIGHT shake to dislodge a FEW bees, and transfer the rest of them directly into the weak hive. Adding TWO frames like this and giving the frames your about to drop in a mist of syrup seems to make them excited and happy rather than put up a loud hum of disapproval.
Drop in a frame of BROOD with no bees on it, the bees dont get excited. Drop in a frame with BEES still on it you will hear the level of the HUM increase dramatically, the mist of syrup turns them to licking it off the newcomers and they are accepted much easier.
NOW, this is what I do. so don't take offense if you do it differently, instead, please post how you do it so I can learn too!