The answer is a bit complex here:
As Paulster2 has mentioned, the battery does help protect from transients, but as the alternator has its own regulator and the computer and audio system have their own as well, this should become useful just if there is something else wrong.
Overall this is not really a problem and the article is from some guy promoting his ebook online...
The issue has to do with with the fact that the alternator is usually not designed to handle a 100% load:
When the engine revs higher, you trigger the spark coil(s) more often, and as you open the throttle, the compressed air/fuel mix is more dense, necessitating more energy to trigger the spark, or when you use the car's audio system power demand increases, or use the windows, etc.
All of these create an extra demand for energy and, during the normal conditions the alternator provides more than enough, but if you suddenly accelerate, open the window and crank the sound up at the same time, it may not and the design expects the battery to pick up the slack since those are temporary conditions.
Not having a battery will create an extra load on the alternator's voltage regulator.
Typically, alternators die from bearing failure, but the second cause of death is regulators. Running the car a little bit without a battery is not a big deal, but running it for long period, even with a bad battery, will wear out the regulator as it will not be able to cool itself sufficiently and will eventually break.