Creating an older installer on a newer Mac

You can create the bootable flash on the new Mac: Create a bootable installer for macOS - Apple Support

Big Sur is available to be created ... page down to the command.

Follow all instructions precisely skipping nothing that applies to you.

Interesting ... I can't get Big Sur to work wither even though it has in the past.

I suggest you try the app Install Disk Creator. I've used it many times and it works perfectly.

I'm as baffled as you are now. Other's will undoubtedly tell us what we're overlooking. I blame it all on old age ...

Can't get Ventura to work either and I know it works.

<<. And no, it did not have a recovery drive set up either. >>

Most Macs after 2011 models have Recovery in their ROM. It CAN'T be erased

Mac Pro 2013 has Recovery in its ROM. it does not requires a USB-Stick installer. It does not have the complexity of a T2 chip (added for the 2018 and later intel models.)

Launch recovery by holding Command-R at startup, and choose to re-install MacOS. it should download and install without incident, provided you have a drive on which it can be installed.

if you don't have a suitable DRIVE, you need to discuss preparing one with Disk Utility. If you have replaced the original internal SSD drive, that could make things MUCH more complex.