It isn’t going to be anything too far out, because doing so would expose some compatibility issues. Lightspeed give these numbers
and googling around suggests that PC headsets are in the same ballpark.
So it should be possible to wire an aviation headset directly to a PC headset output.
Not surprising you can’t find data on the speaker impedance – Bose famously do not publish specifications for any of their products.
It is tied in with the “this is better in ways that you cannot measure” philosophy of the company and its founder.
Have you tried a BT connection – no idea if it would work.
A standard 3.5mm headset output socket doesn’t output any “power”. It is just the audio signal. A 3.5mm mike socket is a different thing; there is a very low power supply available there for powering the mike.
As to why one can’t power an A20 directly from a headset socket, is a good question. Perhaps the A20 needs more power, or perhaps the battery is present to provide power for the headset mike. I have done tons of audio design but not played around with aviation headsets 
I’ve had a dig around and can’t find any data on the headset (speaker) impedance. There is a “low imedance” version being advertised but that refers to the mike, and is not going to work in most GA installations. Maybe @wigglyamp knows.
I’m currently searching for a way to connect my Bose A20 to my PC, to make my experience in Flight Simulator more authentic.
I ran across this adapter:
What I don’t like about it is that it requires a 9V battery. PC headset slots already supply energy, so why is this necessary? Anyone know of a way to connect an aviation headset to the PC without using additional batteries?
I found one second hand (the Flight Sound X 3.0). Works really well. Thanks for the tip.
I have this one:
https://flightsounds.com/collections/solo-aviation-headset-usb-adapter
Works like a charm.
I don’t know, you can try without. Worse case scenario the audio will be lower than usual.
That’s a good idea, I could try that. I was wondering if I needed some power appliance like in the first example to supply the headset(?) with power?
Buy one of these and solder some new jacks?
