Just finished the Penn State course, and their description moves the queen to the new box.
If your colony is preparing to swarm, they will be cutting down the queens rations so she loses weight and can fly and stuff like that. The reasoning for leaving her in the old box might simply to not add more stress to her. If they aren't planning on swarming, then leaving her behind will be "business as usual" with a feeling like there is now more room. But she should continue to lay as usual.
But yes, moving her (false swarm) should make her and the bees that went with her feel like they have swarmed already. As for the bees left behind if they have queen cells already or they try to make them, hope is that they have eggs or very early larva to make a queen.
I think the problems come with locations. If you move the queen (physically) the foragers will be coming back to their old (location) that night, and you will pretty much be left with the queen, her nurse bees, house bees, and whatever hatches from the comb you gave her. If there are not resources in what you moved with her, there will be little coming in, until nurse/house bees promote to foragers.
If you leave the queen behind, the foragers will still be bringing in resources, the nurse/house bees will still be tending brood, and building comb, and foragers comes home at night. The moved brood and their nurse bees will have only their nurse/house bees to rise brood, build comb and create the new queen cells. If not requeened otherwise, it will take some time until she hatches, takes flight and comes back to lay eggs. Mean time, your house bees have to promote to foraging and start bringing in resources, nurse bees have to
promote to house bees and start drawing comb (for the new foragers to bring back stored to fill it) and new brood has to hatch to become nurse bees. You now have a broken brood cycle in the new split so they can be cleansed from varoa etc. You never got to break the brood cycle in the original.
I can see lots of reasons to move her with the new split, but arguments could be made for keeping her in the original. I think the "rule of thumb" is that she goes with the split, mostly because, you want a new queen in your original hive.