Why do boiling points of alkanes and alcohols become closer with increased chain length
It is simple. Longer the chains higher the similarity between them. In other words the OH group are getting "diluted".
It is more difficult for the donor to find the accepting counterpart, and the number density of the hydrogen bonds, both for volume and mass, gets lower.
The interaction between molecules becomes dominated by VdW forces. 

To visualize this think of methanol and of a very very long primary alcohol. In the second case it is somehow "more difficult" to spot the difference versus the parent alkane.