Why do the Noble Truths talk about 'craving', instead of about 'attachment'? Further to these comments, why do the Noble Truths talk about 'craving', specifically, instead of about 'attachment'?

Wouldn't they make as much sense, or even make more sense, if they identified attachment rather than craving as a condition for the types of suffering which are mentioned in the first noble truth?
Craving (lit. "thirst", tanha) is a technical term that means that very phenomenological moment when you (here!) daydream about something (over there!). Buddha says "unsatisfied craving IS dukkha" (suffering) - note how he does not say "craving is the cause of suffering", he says "unsatisfied craving IS". That's because dukkha isn't exactly "suffering", it should be more precisely translated as "unsatisfactoriness", or, as I sometimes translate it, "the feeling of wrongness". So as you see, to say that "unsatisfied craving IS the feeling of wrongness" is almost a truism, a statement that is obviously and self-evidently true. This way Buddha not as much identifies the cause of suffering, as illustrates the nature of suffering, brings it into light and shows it upclose so we can see it for what it is. Indeed, suffering is nothing else but the painful experience of the unsatisfied craving for another experience.

As for "attachment", I feel like this is a long-standing misunderstanding that crept in since the earliest days of Buddhism, when the term upatthana/upattana (to attend) got mixed up with upadana (fuel). In my opinion, the sequence here is that what one craves for, one attends to, thereby fueling the craving. In other words, as you keep obsessing over the object of your craving, you only make it worse.

This makes it clear that upattana/upadana (to-attend/to-fuel, usually translated as clinging or attachment) is not really the direct cause of suffering, but only a sustaining factor. The direct cause of suffering is craving. 

Four Noble Truths (at leat the first three) are focused on the immediate experience of suffering, as it occurs and can be ceased in the present moment. That's why they talk about craving being the cause.

The Twelve Nidanas cover the big picture: how entire experience of "I" and "The World" arises. In that context it's more appropriate to talk about Upadana being a fuel for Bhava, and Avidya being the root cause of it all.  Avidya (invalid/confused understanding ~of the experience) causes reification of experience as external objects which then become the object of craving. This craving causes dukkha, sure, but more importantly for the big picture it leads to ego-centric imagination (Upadana) which leads to individuation (Bhava). 

I hope this analysis of the terms and how they relate to the phenomenology of suffering clarifies the topic.