How can I make a mac bootable usb installer for old macOS versions?

See How to download and install macOS - Apple Support

Unless it is Sierra or older (DMG download) you can only download an OS version that can be run on the machine

doing the download. You will need to do the download on a machine that can run High Sierra. One you have that you can follow the instructions here Create a bootable installer for macOS - Apple Support

The alternative is to at least get something onto the old Mac so you can then download other compatible OS versions and then install the one you want. See How to start up from macOS Recovery - Apple Support and How to reinstall macOS - Apple Support

You can use a compatible older Mac to make a mac bootable usb installer for old versions such as macOS High Sierra. Download the official installer, connect a USB drive (16 GB or larger), and open Terminal to run this command, replacing MyVolume with your USB drive’s name:

sudo /Applications/Install\ macOS\ High\ Sierra.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume

Once the process finishes, connect the USB to your target Mac and start it using the appropriate startup key. Back up your USB data first, as the process erases the drive.

This is a practical way to make a mac bootable usb installer for old versions, provided the macOS version supports your target Mac.

Try Internet Recovery via Option-Command-R boot. If it lands on High Sierra, use workaround #3:

https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/

The InstallAssistant.pkg extraction method can help make a mac bootable usb installer for old versions such as macOS High Sierra. I used this approach because the App Store refused to provide the older installer on my newer Mac.

The process is simple: obtain the installer package, extract the full macOS installer application, and use Apple's built-in tools to create the USB installer. It is completely free and avoids relying on paid utilities.

For my situation, this method was useful because I could prepare the installer on a newer Mac and use the USB drive to reinstall macOS on an older Mac with a replacement SSD. Before trying to make a mac bootable usb installer for old versions, make sure the installer is compatible with your Mac model and comes from a trustworthy source.