@voidspace if your apple ID has downloaded previous versions of macOS, they should still be available in the Mac App Store under the purchases tab. There are actually 2-3 different recovery modes that will do different installers. One version will reinstall the current OS, and another will install the OS the machine shipped with. I believe you need an internet connection either way. The different recovery modes are cmd + R and another key combo, like shift, ctrl and/or alt.
@voidspace apologies for the delay but I finally figured this out. Probably way too pate for your use case, but go to https://support.apple.com/en-us/HT201372. This links to other support pages that have direct download links for El Capitan thru Catalina. This requires no authentication