Why is glycerol the backbone of fat?
Chance should always be high on our list of explanations of why the history of life turned out in some specific way. As James A. Coffman has described, there are a couple of different notions of chance. Evolutionary biochemistry is quite difficult because today's diversity of biochemical genotypes and phenotypes are a biased representation of the ancient past.

However, I think a case can be made that it was not 'pure' chance. 

As your question itself suggests, the chemistry requires alcohols for the esterification with fatty acids to form acyl glycerols. I think that is 'sufficient' to say it is not 'pure' chance that among molecules that it was glycerol, but since you're also questioning "why glycerol?" over other alcohols, let's look at what glycerol has going for it compared to other alcohols.


Glycerol is relatively small. All else being equal, fulfilling a biological function with a smaller molecule requires less energy than larger ones. Compared to polyphenols, for example, this is a no-brainer because (1) building polyphenol would be expensive energetically, and (2) would be difficult to transport.
Glycerol is flexible. The two outer alcohols have the ability to move with respect to the central carbon so that as we go from glycerol to mono-acylglycerol, di-acylglycerol, and finally triacylglycerol, the outer groups' motion can reduce steric hinderance.
Glycerol can be converted into a lot of other things that form a part of metabolism. While it is conceivable that an extremely different metabolism without glycerol could have come to be, the fact that glycerol is directly and indirectly convertable into a large number of molecules with structural and energetic functions does give it better candidacy than molecules that are relatively unreactive or require large numbers of reactions to build anything useful biologically.
While glycerol doesn't have the greatest number of alcohol groups for its size (size in terms of carbon backbone, not necessarily molecular weight), it doesn't have the least number either. Steric hinderance or difficulty to synthesize of alcohols with the same size would make glycerol a better choice.


Hopefully that convinces you that glycerol was not completely random with respect to all other molecules or even all other alcohols. That is, not all the possibilities in those sets are equally good for the biochemical role that we now see glycerol fulfilling, and glycerol is defintely not the worst. However, there may have been other subsets of alcohols/molecules in existence at that time for which we don't have specific reasons for glycerol being better.

In conclusion, no, not 'purely' random.



UPDATE

Reading my answer above, I can see why someone reading my list of glycerol traits may think I am making a just so story on why it is specifically glycerol that we have today fulfilling a biochemical role of joining together fatty acids via acylation. To be clear, I am not addressing 'why' glycerol is the molecule fulfilling this role but rather whether it is purely random that glycerol is the molecule fulfilling this role. In my vernacular, 'purely random' implies a uniform probability distribution, which is to say that in this context each molecule was as likely as any other to fulfill the role that we see glycerol fulfilling in biochemical systems today. By citing reasons why some molecules are better suited in their properties, we show that the distribution cannot be uniform. We don't need to even place where glycerol sits on this distribution to determine that it is not purely random. We only need non-uniformity. The fact that glycerol has traits that make it more likely than others is an additional suggestion that among the molecules under consideration, glycerol was not the least likely.