So, using phosphoric acid results in an insignificant amount of phosphate compared to what's already in the mash. Seems like this would have no flavor impact on the finished beer. What's the advantage of using slaked lime to remove bicarbonates instead? Seems like a lot of work (precipitating the chalk overnight, racking off of it) for a process that also removes Calcium.
Slaked Lime actually significantly increases water alkalinity initially. It is added to increase the pH of the water to above 10 where calcium carbonate becomes insoluble. Increasing the pH to above 11 will also reduce the solubility of magnesium. That undesirable hardness precipitates out of solution and the clear water is decanted off the sediment.
That clear water still has significant alkalinity. Either CO2 is bubbled through the water to help reduce the water pH (it adds carbonic acid) or an acid such as hydrochloric, sulfuric, phosphoric, or lactic is added to bring the pH (and alkalinity) to a reasonable level.
This process is only partially effective at removing calcium or magnesium. If the water had mostly temporary hardness, you can typically only bring the Ca to 30 ppm and the Mg to 10 ppm with Lime softening. If it has a lot or permanent hardness, then the Ca and Mg will be higher.
Phosphororic acid is relatively tasteless in beer for the reason you cite.
Unlike Gordon, I am interested in both engineering water and beer. In both cases, they are my profession.