The internet says swirling is fine, but not stirring. Why is that so?
Actually, I think "the Internet" is wrong on this one, assuming that we are talking about proper caramel sauce.
In most of candy making, you are very careful of crystalization. You are working with a supersaturated sugar solution, and it is looking for the slightest excuse to precipitate. Stirring will clump the sugar out of the solution into crystals.
Some candy types need to be perfectly smooth, while others (like fudge) get their characteristic texture from careful management of crystal size. You shouldn't be stirring there at all.
But once you have reached caramelization, you can stir. What you have in the pan is no longer a supersaturated sugar solution, but caramelization products mixed in a less-concentrated sugar solution. In other words, you have caramel, which is a substance quite different from sugar syrup. And it does not clump into crystals. It is amorphous in its structure, not a crystal, and it's actually got some viscosity (if you leave a clump a solid caramel around for years, it will flow a few centimeters).
And you are not just dealing with pure caramel, but with caramel sauce, which also has lots of liquid added in the form of cream. So much liquid would have also prevented the sugar syrup from crystalizing if it had been added earlier.
So, to summarize, you shouldn't stir a sugar solution during candy making, but you can stir both caramel and caramel sauce.
It seems that somebody learned the rule about hot sugar solutions and decided that it applies in all kinds of candy making, without exceptions. But in fact, it doesn't apply to caramel sauce.