I don't think taking supers off will cause swarming. I have single deeps and multiple supers. I first apply light smoke, use a fume board, and then remove the entire super or each frame and brush off with a brush. You do have to be careful to cover the supers you mevove so as not to cause robbing.
I'm no expert and struggle a lot with swarming myself, but I think for the most part the kind of "space" they care about is space for the queen to lay. Secondary is space near the brood area, especially just above it, where they feel they need to have honey stored before they can swarm. Physical space for the bees is far less important.
Maybe it could be a slight issue if a huge colony was compressed into a single deep, but I don't think a single queen can produce more bees than can fit in 3 deeps, so I don't think removing capped honey supers above that 3rd box would have any effect.
When you see them bearding that's usually more about temperature regulation than physically running out of space, I think.
Look at Ian Steppler's videos. By this time of year he's reduced huge stacks of deeps down to a single deep, causing tons of bearding, and from what I can tell very little swarming. See here for example. He has more of an issue with swarming during the early flow. But I don't know, your climate and bees and management are probably different.
I always have a good number of bees in the honey boxes. Though my hives are using a lot more than a deep for brood, , if I reduced them to a single deep right now, I would see swarming in a week. Even if I just compressed them down to the 3-4 mediums of brood nest they have, They would swarm before the month was done. Good fall flow going on. I am not sure if that would be the same in your location.
If you can, remove, extract, and put wet supers back late in the day within a day or two. Wet supers seem to give encouragement to re filling. They will clean out the wet supers after a few days and you can remove what you need to store them.
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