No... I MIGHT find three headsets in the cockpit. Often times, there are only two and once I only found one. I'm sure you're asking yourself "how can that be?" . More than once, we have taken a delay at the gate and one time we had to return to the gate after not getting a headset to work properly on the way out. Yes..no backup on board. Wasted 45 minutes.

So, if we do like you advocate, we drop a headset in our bag. Shouldn't you now write up in the logbook a missing headset so MX will replace it?
And, if asked where it went, what are you going to say?

I know your worried about germs, I get it. With that in mind, don't you sanitize you own headset? It certainly can pick up all sorts of germs along the course of the day. So, if you SHOULD sanitize your own headset prior to flight, for OPTIMUM germ prevention, then (as Hillary famously said), what difference at this point does it make?

As an aside, last fall I decided to carry with me a supply of those Clorox/Lysol soaked wipes. Works great. Not only can I amply sanitize the headset, but the O2 mask and the cockpit nest in general. For the first time in recent memory, I didn't catch the flu or even a cold over winter. Germ problem solved and now I have one less tangled electrical cord to lug around. I rather like it that way.