The recipe's goal is to use the natural bacteria in the beets and beet peels to cause the fermentation. The thing about using natural bacteria is that you never know what these are going to be, so it's a crapshoot what flavors you are going to get. Beets produced in one farm may have radically different bacteria in them, different varieties grown in the same patch of land could also vary significantly. It's the same process as making a sourdough bread starter using natural yeasts - you may get a nice smooth and delicate flavor, or you could get something that will take the paint off of a door. The point is that you may be doing everything right, it's just that's what you are going to get from the process.
Looking at the science of the natural yeasts and assuming the recipe writer is not talking complete BS, it could be that the delicate bacteria spoken of in the recipe is being out-competed by the other natural bacteria in the water you use, and on the surface of the container, lid, and airlock. The thing to do in this case is to reduce or eliminate these other bacteria so that the beet bacteria will be the only game in town. To do this use sterilized water (buy distilled or simply boil your water for a few minutes before letting it cool completely), or run your water through a filter which will remove or kill bacteria. Then use a chlorine solution to sterilize every single piece of equipment you will use. That means the container, lid, lock, spoons - anything that will come into contact must be very clean. Wash your beets in sterilized water as well. I'd do that to the equipment even if I used a culture, it will increase my chances of a good result.
If that doesn't work you may have to accept that is the way natural kvass will taste. It could be that the writer of the article likes it that way, or simply doesn't know what he/she is talking about. In that case consider using a culture - there's no dishonor in it and you'll get something drinkable.