I agree it is a difficult issue to get your head around, especially since there are so many different terms, words and definitions involved.
You just introduced a new one (that is: new to this discussion): 'biological fitness'. Now you define it as the ability to produce offspring.
Then you go on to say that a diverse gene pool is facilitated by increased biological fitness.

So, if a population has an increased number of offspring each generation, there is a larger gene pool and a larger population is more likely to survive a calamity.

But I'm having trouble picturing this. On the one hand you describe mechanisms that increase diversity in the gene pool (recombination and mutation, of which officially only the second does increase the diversity, since the gene pool is not only what is actually there, but also what is potentially there, but that aside). Next you say that these traits and combinations have good, bad or neutral consequences for (future) biological fitness.

So far so good, but here you seem to skip a step, for when it comes to NS, doesn't that select certain traits to live and others to die out? If not, it's not really selecting. So on the one hand mutation (and if you insist: recombination) may increase variation in the gene pool, doesn't NS decrease variation, leaving only the ones that give the most biological fitness (or eliminating those with the least biological fitness)?

This confuses me, since your next statement is: "given a diverse gene pool (which is facilitated by increased biological fitness)"
while I thought that biological fitness was limiting the diversity of the gene pool. At least at first sight.

And there still is the possibility of a tautology when NS is defined as "those with a higher biological fitness have a higher probability to contribute to the gene pool". Because giving a greater contribution in the gene pool is the definition of having a higher biological fitness. NS then describes an observation, while it pretends to explain one.