Simulating DNA sequence evolution in R
What are you expecting these lines to do?

num.transi &lt;- rbinom(n = 1, size = length.seqs, prob = transi.rate)
...
num.transv &lt;- rbinom(n = 1, size = length.seqs, prob = transv.rate)


Using the variables that are specified in your code, they are extremely zero-skewed. This is not too surprising given the mutation rate and sequence length.

I expect that increasing the sequence length, or increasing the transition rate, should result in more haplotypes being generated.

After running this through myself, I see that increasing the mutation rate does increase the number of haplotypes, but the skew is retained. This is because each mutation creates a new haplotype, following something like an infinite allele model.

I guess this is not particularly surprising. It's unlikely that two random mutations from a particular haplotype will result in the same haplotype, and therefore any mutations (from any existing haplotype) will generally create something new.