Discrepancy in time for genetic differences between human and chimpanzee to accumulate
Good literature work here and good question +1! In short, your main mistake was that you based your calculations on a single site and not on the whole genome. More info below.

Genome-wide vs sites specific mutation rate

The statistic of $10^8$ generations that you computed is the average rate of fixation of new neutral mutations per site. As considered the genome-wide statistic between humans and chimpanzee, you have to scale your predictions up to the entire genome. There are about $3 \cdot 10^{9}$ nucleotides in either the human of the chimp genome. That would result in a genome-wide mutation rate of $U ≈ 2 * 10^9 \cdot 10^{-8} = 20$  (the two is because mammals are diploids). Hence, the rate of fixation of 0.05 generation.

So, the expected time it would take for $30 \cdot 10^6$ new mutations to fix would be $30 \cdot 10^6 \cdot 0.05 = 15 \cdot 10^5$ generations. Because, such mutation can happen in either lineage (humans or chimps), we have to divide this number of 2 leading to $7.5 \cdot 10^5$. With a generation time of 20 years, that would take $1.5 \cdot 10^7$ years or 15 million years.

15 million years? Why not 6-8 million years as expected from fossil records?

Most of the discrepancy is probably due to vague approximation made in the computation. For example, you assumed a mutation rate of $10^{-8}$ while in reality, the average mutation rate is rather twice as large (Nachman et al., 2000, Kong et al., 2012, Wang et al., 2012, Rahbari et al. 2016).

Doubling this mutation rate would divide by 2 our expectation of 15 million years leading to an expectation of 7.5 million years that perfectly match the fossil records!

There is however another source that may cause an overestimation of the time needed... You assumed the absence of incomplete lineage sorting or in similar words, you assumed that the standing genetic variance in the ancestral population negligibly affected divergence among the two lineages. However, these discussions are for another time!