I would actually get this $10 dongle just in case anyway, it can serve as a reference to check whether things are okay. You don't have to use it if you don't think it improves anything, but it is a trivial investment to answer a question you might not be able to answer otherwise. Based on the fact that you have 560S with a pretty high impedance measurement of 133 to 224 ohms (with nominal given as 120 ohms) I think any device with minimal attention to quality ought to be able to drive this headset correctly, but perhaps you want to buy other headsets one beautiful day.

The Apple dongle has been measured and is known to be an accurate DAC+amp combo, with 0.5 (EU) or 1 V (US) maximum output (so it can't drive high impedance and low sensitivity devices that typically want higher voltages than that), but it has very low 0.9 ohm output impedance so it can correct for frequency response errors that occur when high output impedance drives impedance load that varies by sound frequency.

The error comes from voltage drop that occurs somewhere in the output plug's electronics where voltage drops below expected because the headset wants more current than the output stage is capable of producing due to its internal resistance. There is a rule of thumb that says that output impedance should be at most 1/8th of the lowest impedance of the headset, e.g. if headset is 120 ohms, then you should have at most 15 ohm output impedance (which I daresay is an easy requirement, though some mobos have been measured to have more than that). This should limit errors related to output impedance to level that is probably not audible, to a small fraction of a dB.

There are barely any headsets with impedance less than 10 ohm. I happen to have Crinacle x Truthear Zero which actually has 10-20 ohm impedance in such a way that it has a crossover internally where 20 ohm impedance is with the bass and 10 ohms otherwise, and it benefits from low output impedance amplifier or the bass increases too much. So I use the Apple USB-C dongle for those because otherwise it becomes pretty bass-heavy in one laptop, though it is actually fine in another.