Hey, where did your comment go?
Anyway, in case it helps anyone else - I did as you suggested and bought this ANDUL USB Switch Extension Cable for $7 on Amazon. It works just fine. It's an actual hardware switch, which works good with my setup.
Yes, if I switch it off, my PC speakers come on. Just like they would if I unplugged the headphones. I can hear and talk just fine.
I mounted it to the front of my PC case with some double-sided 3M tape. Now I just reach down and click it when I want to switch between headset and speakers.
That's not a bad setup. I used a usb switch with data passthrough to flip it on and off on our Ioniq 5 so I didn't wear out the port by unplugging it all the time. I might change my setup to this so it's fully automatic.