Creating Mac OS USB Installation Media on an Apple Silicon Machine

A 2014 Mac Mini originally shipped with macOS 10.10 Yosemite although units purchased later on shipped with macOS 10.11 El Capitan.

This means if you have access to another working Mac generally from 2007 to mid-2022 (Intel or M-series), then you can use it to create a bootable macOS USB installer. The version of macOS depends on the exact model of the other Mac (the other Mac needs to be compatible with the OS you want to create). Generally these models suffice for:

• macOS 12.x model from 2015-mid2022
• macOS 10.15 model from 2012 - mid-2020
• macOS 10.13 model from Late-2009 to mid-2018
• macOS 10.11 model from 2007 to 2015

You can confirm compatibility using the information in the following article:

https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

FYI, if you installed a 2.5" SATA III SSD, then you can boot using any version of macOS 10.10 to 12.x. However, if you installed a third party PCIe NVMe based blade style SSD, then you are limited to macOS 10.13+.

As others have mentioned, using Command + Option + R is the best method to attempt to access the online Monterey installer. Unfortunately some Macs may only boot to the online installer for the OS which originally shipped on the computer from the factory regardless of the keys used.

If you are trying to download the macOS 10.10 - 10.12 installer as a DMG, then you need to extract the real installer using a Mac compatible with that OS. The .pkg file within the mounted DMG won't run on an unsupported Mac. This is the only place to download the full macOS installers (any other DMG OS files you may find on Apple's website are most likely update patches & not the full installer):

How to download and install macOS - Apple Support

How to download and install macOS - Apple Support

explains how to create your own installer.

To avoid permissions issues, you need to make sure the files you are migrating over are migrating to the same or newer OS and find themselves in the same exact folder path down to the drive name.

And when you create a new user on a blank drive, make sure it has the same passwords for login as original.

I shall leave you with this ditty and retire from the question

You are very well supported by Both the Other 2 Contributors moving forwards

To perform this action will require booting from a Bootable Installer. 

The Bootable Installer can Only be performed on an Apple Computer - Full Stop

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Monterey would have to be done on a computer that Qualifies to run Monterey.

Notation: If the computer being used to perform this action is Too New or Too Old to qualify to run the version of macOS - this computer can’t be used.

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend or associate.

Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

Choose the Upper Most Drive ( not the volumes indented and list below ).

The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map. This applies to macOS 10.14 Mojave and above. 

Once that is done >> backup out of Disk Utilities and choose install macOS. 

Follow the prompts and it may automatically reboot several time. 

Upon a final reboot - Setup Assist will present with the newer version of macOS.

You can use Setup Asset and point it to the Time Machine Backup Drive. This is a One Time offer. Thereafter you will need to use Migration Assist 

Not really

The big problem is that Sierra does not support APFS.

September 24, 2017 is the newest Mac you can install Sierra on or build a Sierra image from.

Now your Late 2014 Mac Mini does support

High Sierra, Mojave - the newest 32 bit application operating systems from 2017 and 2018. If your applications are fall northern Hemisphere 2018 or older, you'll probably want to double check if they are 32 bit only.

And your Mac supports Catalina, and Big Sur, which are 64 bit only.

So if you have a Mac from early 2020 (Intel), or earlier you can create an image that can be used with the Mac Mini.

Option-Command-R: When you press and hold these three keys at startup, Recovery might offer the latest macOS that is compatible with your Mac.

The operative wording is >> Might