I have been working with ESRI on a fix for sometime, and it appears that they have found a solution. The fix is to use Cell Statistics, ensuring that NoData is not considered in the calculation. As many people may have discovered, Cell Statistics really only creates a pseudo-intersection of values. In order to correct the problem, one needs to create a raster with the value of zero, which embodies the full spatial extent of all rasters for which the "merging" needs to happen. Then, that raster, and all others need to be included in the Cell Statistics tool, using the option Sum values.
It does work, but it can also be quite slow depending upon size. If others have similar problems and find that this solves the problem, please report your successes/issues here, so we can all benefit!