I’m chief apple basher but fair play to them.
Its a grand sting to get you to buy another computer – a sympathy buy, after all they’ve done absolutely everything possible to help you 😉
As an aside Carbon Copy and Super Duper are very slow/inefficient way of doing a back up and I’ve had them fail numerous times. Time Machine is great and if you ever need files back or a complete restore (eg when putting in a new drive) it works really well. What I should do is get N offsite backup in the event of a house fire
OP Time machine copies your music/photos. In fact if your whole disc or computer dies it will recreate everything inc the operating system