You're hosed if you're switching focus away from Zoom (i.e. presenting). The keyboard shortcuts (Alt-A or holding-space-to-unmute) only work when Zoom is in focus

I have 2 monitors, and drag the zoom controls to the non-presenting screen

Getting a headset with a physical mic is probably your best bet

Side note: noise-cancelling headphones are way more useful than you'd think. I delayed for 5 months before buying them, and regret every minute I didn't do it earlier. Doesn't help with the mic, but I didn't realize how much extraneous household noises contributed to my malaise.