From the same article:The problem with allowing the CO2 reaction to occur as you mix your Kalk in a container without Vinegar is that you are mixing it in a comparatively small quantity of water that contains only a very limited amount of CO2. So, instead of having enough CO2 to go all the way to useful Bicarbonate ions (HCO3-, or Alkalinity) as you want it to, the chemical reaction stops at the undesirable Carbonate ion (CO3--). This is what sets the stage for the Calcium ions to precipitate out as useless solid Calcium Carbonate while still in the mixing container -- the Kalk mixes with SOME CO2, but not with ENOUGH CO2.
If your mixing container contained ample CO2, i.e., enough to fully react all the Kalkwasser as Bicarbonate ions, you would indeed then want to do all the mixing with CO2 right there in the container. It's only because there's not enough CO2 present in the small container that you need to limit the mixing to avoid precipitation.
So, you've actually hit on one good reason WHY we want to add the Vinegar to the mixing container -- the Acetic Acid in the Vinegar provides the equivalent of enough CO2 to take the reaction all the way to the desired end products for which we drip Kalk -- that is, to produce free Calcium ions and free Bicarbonate ions in solution in our tanks.
By using enough Vinegar, we can mix to our heart's content right in the container, and don't need to worry as much about dripping the solution slowly into the tank. We have already driven the reaction all the way to the desired outcome, and provided ample CO2 to produce a balance of free Calcium ions and free Bicarbonate ions. This, incidentally, is why Kalk is said to be "ionically balanced" - it produces a one-to-one balance between Calcium and Alkalinity.
By the way, the "stoichiometric" amount of Vinegar, i.e., that amount that provides the exact equivalent of enough CO2 to react all the Kalk powder to Calcium and Bicarbonate, turns out to be about 25ml of 5% Acetic Acid per liter of saturated (0.02 moles/liter or 1.5 grams/liter) aqueous Calcium Hydroxide solution (Kalkwasser). I've used 30ml of Vinegar to a ½ teaspoon of Ca(OH)2 per liter of mix without any problem, but recommend about 15ml to those new to using Vinegar. This means that you still need to go easy on the stirring, because we are not providing quite enough equivalent CO2 to avoid SOME Carbonate ion formation if we get carried away with the mixing.