You are assuming (as did Kai,apparently) that if you can't see it it hasn't precipitated. When I first really figured out what was going on I had dissolved chalk in acid adding acid until the pH was well below neutral and the solution crystal clear (or so I thought). But the amount of acid added was not stoichimetric with the amount of chalk dissolved and the pH kept climbing. I just left things alone to see what happened and found that the pH was still rising the next day. Clearly the reaction was not complete. It is the fact that a solution made in the way that you advocate continues to contribute alkalinity for 24 hours that concerns me. You add enough to get your mash pH but those microcrystals keep absorbing protons for days raising your kettle and fermenter pH. There are better ways to get some alkalinity into mash than carbonate.
If you read that paper I don't think you will find this method discussed or advocated. But it has been a long time since I wrote it. I freely admit that I now know a lot more about this subject than when I wrote that paper.
Ca++ + 2HCO3- <--> CaCO3 + H2O + CO2
The CO2 that leaves comes from the decomposition of the two bicarbonates into carbonic acid (which in turn quickly decomposes into water and CO2) and a carbonate ion which is quiclkly grabbed by a calcium ion and precipitated. This if CO2 leaves the solution CaCO3 has to as well. The statement is thus incorrect. What happens is that microcrystals form which would require a microscope to see.
At higher temperature equilibrium is reached pretty quickly. You have made water super saturated both with respect to CaCO3 and CO2. Why do you think it will stay in equilibrium - especially when heated and stirred? Note: The alkalinity of water in equilibrium with the atmosphere over chalk is about 1 mEq/L.
I guess I should also note that at one time I thought this was the answer to a lot of brewing problems and even considered setting up a garage business to make calcium bicarbonate solution for sale to home brewers. Then reality set it.