You may be missing an important implication, but you also may be missing something a bit simpler.
As they were happy, there was something missing.
In this version of the sentence, we're given nothing about the degree of happiness. We're given the fact of their happiness (without any amount) with the implication that that's the reason that something is missing. Explicitly, happiness and the absence of an unidentified thing are correlated and concurrent -- they happened to happen at the same time. Still, this version is likely to be understood as something similar to "because they were happy".
Given the right context, it might not seem strange at all that their happiness causes a lack of something else. Without that further context, this model sentence seems strange. The natural implication of the original model is missing here.
[[Even] as] happy as they were, there was something missing.
In this version, we are given something about the degree of happiness. It's notable and remarkable, as the degree itself is what's been noted and remarked upon. Here, their happiness is not the reason that something was missing. Instead, their degree of happiness stands in contrast to that fact.
The contrast could be explicitly marked. We could paraphrase the sentiment with phrasing like "despite how happy they were" or "although they were quite happy" or even (as noted in the brackets above) "even as happy as they were".
Changing the word order changes the grammar. With "happy as they were", the prepositional phrase modifies "happy". In effect, because it stand outside the clause in which it could function as an argument, it remains available to modify something else. With "as they were happy", the "happy" modifies "they" as a subject complement only. At that point it's job is done and it's not available to do any more work.