Variance of periodogram estimate of the power spectrum
There are two sources of error in estimating a quantity: variance and bias.  In your question you have only considered variance.

The improvement in the spectrum estimate using the periodogram comes because there is less smearing of frequency components as you include more samples in your estimate of the power spectrum.  Using a longer data window causes less power from one frequency to be leaked into other frequencies.  However, this only reduces the bias of the estimate.  

So, as you use more samples the bias is decreased, which means you do have a better estimate.  However, the variance remains unchanged, as stated in Numerical Recipes.  A discussion of this is given on this web page:

http://matlab.izmiran.ru/help/toolbox/signal/spectra7.html