But how does the clock of the switch affect the sound quality?
As an IT professional, I know how switches work. The Layer 2 audio switch handles only Ethernet frames.
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What does it mean to reclock packets in the switch? The switch doesn't even recognize IP packets, only Ethernet frames.
Short answer, in only two ways. Spreading EMI (it also interacts with ground loops over cabling system) and spreading own noise through the power supply and power cables.
It is true, reclocking frames does nothing, up to 100 meters, only exceeding such lenght requires a adding switch.
As long as packets arrive in time, there should be no distortions added, it is considered fully asynchronous, a receiver use its own independent clock. If some frame is dropped on the path, there will be a dropout in the sound stream, it will be noticed immediately (not as a sound quality, but a violent disruption).
However, there is additional traffic a receiver has to interpret like truncated frames coming out of the switch or broadcast/multicast traffic, it create peaks of EMI and the power supply noise, see the above. This is why reducing traffic transfered over a cable or FMC is important. Special protocols for audio were developed that reduce bursts in the traffic, so a noise is spread more evenly across wide spectrum (less higher energy peaks).
Why does the audiophile switch need a good clock, what is the advantage of an OXCO clock or an external clock?
This is a question worth of one extra external clock.
Adding external clock cable changes distribution of ground loops, it may have a positive effect on SQ. The same effect as using mysterious ground boxes. If helps, fine. However it make a noise created by a switch correlated to the sound, it is a negative efect. I always suggest: if you want to add external clock to a switch, use a different clock device, not your audio clock for D/A conversion.
Many technologies leak to the audio industry without any sense. In this example enterprise grade networking equipment in datacentres (10Gbps Ethernet and above) use 10MHz external clock, but for a different purpose. It is for token recognition. It is easier to recognise transitions of logic levels when the exact frequency of incoming datastream is known. The exact timing (frequently synchronised with GPS) is also needed to setup a precise time advance in a transmitter for avoiding colisions in multiple access networks like 5G.