Confused about prefiltering environment maps (Manson 16)
Your first quote is referring to "Split-sum approximation" presented in "Real Shading in Unreal Engine 4" by Brian Karis, and also referred in the paper [Kar13]:

$$\frac{1}{N}\sum_{k=1}^N \frac{L_i(l_k)f(l_k,v)cos\theta_{l_k}}{p(l_k,v)}\approx \bigg(\frac{1}{N}\sum_{k=1}^N L_i(l_k)\bigg)\bigg(\frac{1}{N}\sum_{k=1}^N \frac{f(l_k,v)cos\theta_{l_k}}{p(l_k,v)}\bigg)$$

In split-sum approximation the incident radiance (cubemap) is pre-convolved with GGX kernel using for example importance sampling, and normal-incident specular reflectance ($F_0$ or "specular color" you refer to) is multiplied with another pre-convolved 2D look-up-texture (referred as "an auxiliary specular color map") shown below.



Next about your second question of importance sampling being inefficient due to band-limited signal - It simply means that the incident radiance is sampled with finite resolution (cubemap resolution) and thus higher frequencies have been filtered out. But because importance sampling biases samples to important regions of the function (peak of the GGX BRDF) this region may be sampled in unnecessarily high frequency thus wasting samples.