In other words, you said my hard drive would have to be functional to use Time Machine. But, why can't I boot Time Machine off of my external drive if my internal hard drive dies?
You don't boot one drive off of another. You just boot a drive. What he meant was that TM can't restore your data without a functional drive to restore it to.
If you are backing up by cloning an internal drive onto an external, and the internal dies, you can then boot off the external and get back up and running immediately. The disadvantage is that you no longer have a backup; the external that used to be your backup is now your running copy, and if anything happens to it you're SOL.
So, in the event that your internal dies, your first order of business is to buy a replacement drive, and clone your backup onto the replacement. Then you can start using the replacement drive as your boot drive, and continue backing it up as before.
If you back up using Time Machine, and your internal drive dies, your first order of business is to get a replacement drive and restore your data onto it from Time Machine. Then you can start using the replacement drive as your boot drive, and continue backing up as before.
The only difference is that a clone backup gives the option, if you're really in a hurry to get back up and running, to boot off the clone, knowing that doing so leaves you with no backup.
If you can afford the time to wait for the replacement drive to arrive, and the time it takes to restore, both methods are equally effective. Restoring your data to the new replacement is just as fast whether you're restoring it from a clone or from Time Machine.
