Why do (almost) all energy carriers contain adenine?
As @canadianer comments, this question is unanswerable, and it verges on being classified as ‘opinion-based’. However, because I do not find the answer from the OP appealing, I’ve set out a few points of my own. Hardly an answer — more a list of alternatives as food for thought. 

I can imagine adenine being chosen for one of the following reasons (others are welcome to suggest additions):


Functionally best suited
Was there first
Easiest or cheapest to synthesize originally (which would explain why
it was there first, if this were the case)
Random choice


Let’s look at them in turn.

Adenine functionally best suited

There seems nothing in the function of NAD that suggests the adenine ring could not have been replaced by another of the bases, and we know that the free energy of hydrolysis of the beta–gamma phosphodiester bonds of GTP, CTP and UTP can be utilized in an analogous way to that of ATP, so this doesn’t seem very likely. 

However one does wonder if a purine (rather than a pyrimidine) ring conveys some sort of advantage in that the other purine-based triphosphate, GTP, is used so widely in the (ancient and essential) process of protein biosynthesis. Perhaps the greater area of contact of the purine ring favoured interaction with proteins (if you don’t argue that these came later).

Adenine was there first

One might consider that early in the evolution of life the bases (or nucleosides and nucleotides) emerged one at a time, in which case the first might have been used for NAD etc. and stuck. One might also consider whether a two-base world preceded a four-base world. If A and U (but not G and C) were in the first RNA (which may have been single-stranded, so I don’t see that the GC content of contemporary thermophiles is relevant) then it may have been a choice between A and U, which A might have won because U is subject to spontaneous deamination (or for the protein-interaction argument mentioned above).

Adenine as easiest or energetically cheapest to synthesize originally

Clearly, the first base might have been the one favoured by chemistry in relation to its synthesis. Contemporary biosynthesis is unlikely to reflect primeval biosynthesis, but it is amusing to observe that adenosine biosynthesis is the only one that doesn’t involve NAD-dependent oxidation. Of course all the contemporary biosyntheses utilize ribosyl pyrophosphate, which requires ATP in its synthesis from ribose phosphate. (Talk about chicken and egg!)

Random choice that stuck

In the absence of any more sophisticated explanation, then it could just be this. Once a choice was made and enzymes evolved to use NAD etc. the choice of adenine couldn’t be changed (or there was no strong enough functional advantage to doing so). 

Postscript: When and how did NAD appear on the scene?

One of the reasons I don’t favour the OP’s answer is because it envisages NAD arising after DNA genomes and as a separate entity. I would suggest that it is likely that RNA preceded DNA and that catalytic RNA was responsible for some of the early oxido-reductions. (The first ones were probably catalysed by iron–sulphur centres, either inorganic or associated with simple proteins.) It seem feasible that the nicotinamide ring was an elaboration of a ribozyme which was ultimately almost completely replaced by protein. The adenosine was retained with the functional nicotinamide because it could interact better with the protein. Gross speculation, and it doesn’t explain why adenine was the chosen base, but an antidote to thinking NAD evolved in the fully grown form we see it today.