No, there is probably no protocol, e.g. digital audio protocol, that the headset would receive over the analog cable. Also, it is usually not described if the analog audio will be used and sent to amplifier, or whether it needs to be digitized first. Chances are that the amplifier in the headset is class D and it can only work from digital signal (it is both a DAC and amplifier at the same time).

At least noise cancelling headsets usually have wildly different frequency response when you turn them on vs. when you connect analog to driver and drive it directly. I can't quickly find out the answer to this question: does the sound change if the headset is turned on vs. off? In the headsets I have owned that have had optional wired connection, the sound has always been absolutely terrible when powered off because manufacturer actually fully relies on DSP (or whatever amplifier circuit) to fix the frequency response of the driver.