Does Time Machine back up the operating system?

Time Machine actually does backup the macOS along with your user files.

In doing so, however, the Time Machine backup drive itself is not bootable. The backup can be used to restore the Mac to the condition it was in at the time that backup was made IF the Mac can be booted in recovery mode and the "Restore From Time Machine Backup" process is invoked. This will require that a valid recovery partition has been created on the startup disk (usually automatic with a new OS installation) or that the Mac has a working connection to Apple's internet recovery servers using the appropriate startup key combinations.

Starting up the Mac in Recovery mode or Internet Recover mode is done by using the keyboard commands described in this support document –

How to reinstall macOS - Apple Support

Please also see:

How to Fully Restore macOS From a Time Machine Backup in Recovery Mode - howtgeek.com

No, it only backs up your data. The OS is separate, https://support.apple.com/en-us/HT203981. This article states OS needs to be installed before migrating any files from your TM backup.

:) 👍

You are correct that before you can use Migration Assistant the macOS must be installed. 👍🏽

A Time Machine backup can be used in Recovery mode to reinstall the macOS. 🙂