I think the question you posed deserves an official answer. Simply because a crawlspace exists, does not automatically qualify it as a damp location.
Further, if the romex simply passes through the space, even it it is damp, should not necessarily cause a problem. As to damp locations, I believe the restriction is more specific, as to the amount of moisture present. And, the truth of the matter is that there are thousands of homes where this cable is installed in damp crawlspaces. The question is whether the crawlspace is simply damp, or actually wet. I also question the question, in its entirety, as the NEC specifies a normally dry location. I do not believe crawlspaces are constructed, nor acceptable, in a normally wet condition.
I agree that damp locations may be problematic for metallic sheathed cabling, from the standpoint of the long-term affect of the moisture on the sheathing. As we know, some metallic sheathed cables have a separate conductor embedded inside the sheathing for grounding, while others have aluminum tracer wound around the sheathing, which is not affected by rust, and is installed to specifically allow a continuous grounding path via the metallic sheathing. Either scenario is no more affected than EMT will be. EMT will rust in time. For some applications, sealed fittings are required on EMT.
I believe the correct answer is that it depends. It depends on whether the crawlspace is truly considered as a damp location. Was it designed to be normally dry or normally wet, because the latter brings in an entire new set of rules for the inspector. The conditions are determined on-site.
Then, the question as to why the crawlspace is damp or wet raises other questions, like are site drainage issues causing the problem. Is there adequate ventilation in the crawlspace? Why is it wet? Then, are there open j-boxes present, which may prevent the inspector from venturing into the crawlspace? Dirt floor or rat slab? Entry from inside the dwelling or outside?Are the runs in question original, or part of a retrofit? Is there metallic sheathed cabling present, which is rusting or broken?
IMO, barring any applicable, non-obscure ruling, open to interpretation, then the situation needs to be evaluated on a case by case basis.
Indeed, the crawlspace in my home is vented to the inside basement. It is 3’ high, has a concrete rat slab, and is dry. By the way, its loaded with romex, inspected and approved by the Board of Fire Underwriters.
So, in my case, the answer is yes, it is fine. Either romes or bx, or EMT