According to lead rules designer Jeremy Crawford:

The intent is that the shield hand can be used for a somatic component if the holy shield is used to cast the spell.

In a tweet exchange on 21-23 November 2014, @Sword_of_Spirit asked:

@JeremyECrawford Holy symbol engraved shield counts as material component, but does holding the shield replace somatic component also?

Crawford replied:

@Sword_of_Spirit The intent is that the shield hand can be used for a somatic component if the holy shield is used to cast the spell.

And further clarified that the shield arm cannot be restrained:

@Sword_of_Spirit You still need use of the shield arm to move it while performing the somatic component.

Spell descriptions are rules, so this is a case of the specific overriding the general, yes. The spell says you touch thumbs to make the spell happen, so to make the spell happen you touch thumbs.

The alternative—selectively designating some parts of the spell as "real" rules and other parts as not—is entirely doable as a DM's house rule, but as it requires a personal interpretation it has no foundation in the rules as written beyond the permission they give to the DM to create house rules.

Addressing the objections

One common motivation for assuming that there must be a distinction between "ignorable fluff" and the "real rules" is familiarity with game designs that deliberately create such a distinction. That is an idea that isn't part of most editions of D&D, including what we've seen of 5e's core. Such a separation is a prominent feature of 4e, and that might lead one to believe that it naturally exists in all D&Ds or even all RPGs, but separation of game text into "fluff" and "crunch" isn't the default in RPG design, and is particular to the RPGs that feature it.

It's plain that no such distinction exists overtly in 5e spell descriptions: they have no separate "flavour text" or any other distinguishable separation between required and optional parts that can be pointed to and agreed upon, without making an arbitrary decision. All of the descriptions are in terms of what happens, freely mixing mechanical and the fictional effects with no demarcation to say that this is a rule but that can be disregarded. Expecting such a separation, taking it as a given, will motivate one to look for one, but expecting something to be true doesn't make it so.

It has been suggested that the first sentence is the fluff and the rest of the spell crunch. This argument could be discarded swiftly by pointing out that there is nothing in 5e that ever suggests this, and it's invalid to try to interpret 5e based on the conventions and structures of some other game; but this dismissal has more support than that. Looking at three spells chosen randomly, it becomes evident that this proposed division doesn't exist: astral projection has rules for targets and duration in its first sentence; resistance specifies a hard condition for its target ("willing") similarly; ice storm's first sentence is the only place its area of effect is given. Indeed, the first sentence of burning hands is the only place where its area of effect is described as a two-dimensional cone rather than a three-dimensional one, a small but substantial rule that could mean the difference between life and death for friend or foe alike. This proposed line between the first sentence and the rest of the spell description as being the fluff/crunch divide has no merits to raise it above any other arbitrary dividing line, and actually has severe drawbacks for being able to use many spells.

An argument could be marshalled that D&D's recent editions establishes a weight of historical precedent for a fluff/crunch divide that 5e is natural heir to. But if that was a reasonable analysis of 5e's text, it would be present unambiguously in the text. That it is difficult to read such a division into the text despite that prominent heritage is a very strong argument that it doesn't exist in 5e. Further, there is actually a much longer history of the indivisible fusion of fiction and mechanics in D&D. This is the heritage that the designers of 5e have been explicit about drawing from in its design, which makes the argument that a division doesn't exist stronger by establishing the likelihood that it's a deliberate design choice. It's hardly something they could have intended to include, but forgot.

Again, all this can be overridden by house rulings of course, but such a separation is not in the rules or even implied, and in many places strongly contraindicated. We may see such guidance in the DMG as an option, but from a structural analysis of the rules, it is not the base state of affairs—the rules resist attempts to locate a dividing line, as questions like this attest.