I understand your concern about the availability of free software for moving your operating system from one computer to another. While it is true that some software that used to be free may now be paid, there are still options available for cloning or imaging your operating system.
One option for cloning your operating system is to use a live CD or live USB drive with a cloning software such as Clonezilla. This will allow you to boot your computer from the live CD or USB and clone your existing operating system to the new drive.
Another option is to use a tool such as the Microsoft System Image Backup utility, which is included with Windows 10. This tool allows you to create a full system image of your current operating system, including all of your files and settings, which you can then restore to the new drive on your desktop computer.
It is important to note that cloning or imaging your operating system is not the same as syncing your Firefox browser. Syncing your browser simply means that your bookmarks, history, and other settings will be copied from one computer to another, whereas cloning or imaging your operating system will create a complete copy of the entire operating system, including all installed programs and settings.
I hope this information helps. If you have any further questions, please don't hesitate to ask.