The end of a center drill is a spotting drill.
While technically this may be true, this statement strikes me as a bit misleading to those with less experience. The end of a centre drill is way too small and fragile to use for every day spotting of hole locations. Not to mention that if you penetrate the material deeper than the cutting lips of the small point, you've defeated the purpose of spotting/"centre drilling" in the first place.
Does it work? Sure it does. I'll even use a centre drill to spot holes if it isn't too critical and I have one sitting right there beside me.
Would I trust it for anything I wanted to be absolutely "spot" on? Nope.
But I'm digressing horribly from the original topic of this thread. In this instance the best reason not to use a centre drill is that you're making an operation more risky than it needs to be. If you break off the tip of a centre drill occasionally, that's par for the course.
Been there, got the T-shirt. If, however, you break off the end of a spotting drill in a part..... well, worrying about the drill wandering off a few thousandths this way or that is the
least of your problems.
An option I favour that hasn't yet been mentioned is to use "stub length" or "screw machine" drills. They are much, much more rigid than their "jobber" counterparts. Most of the time you aren't drilling more than a few times the diameter anyway so the extra length is just extra length. They're stupidly expensive to purchase, but making your own set by cutting a regular length set is excellent practice in drill bit sharpening. With care, properly sharpened stub drills can be used on their own for all but the most critically located hole.