Booting from External Hard Drive in Catalina

I have a 16" MacBook Pro purchased in the summer of 2020. It's running Catalina 10.15.7. Apple repaired it, replacing the logic board, so it's just an OS.

I previously asked if I could run my laptop on an external hard drive running Time Machine, and I was told NO. So I just used Carbon Copy Cloner to copy my hard drive to an external hard drive. But when I called Apple Support just now, they said Time Machine CAN be used to boot a computer, and it's the ONLY method they support.

Anyway, I've copied my data back to the internal drive in the past by rebooting my laptop, holding down the option key, then choosing the external hard drive as the startup disk. I can then open Carbon Copy Cloner (CCC) and copy my external hard drive to my internal hard drive.

Apparently, something changed in Catalina. When I used that method, I don't see any options aside from my internal hard drive.

I've found several sets of instructions online, all of them confusing. I tried restarting my computer in Recovery Mode, then selecting my external hard drive as the startup disk, but that didn't work.

So what is the magic solution? All I want to do is clone my external hard drive to my internal hard drive, the way I used to do.

Thanks.