Well, Crawford is an expert on Old Norse language but not necessarily on other aspects, nor does he seem to always do proper research as I've seen a video of his repeat false claims straight from Wikipedia articles.

Anyway, there are no Norse primary sources from the Viking Age other than short runic inscriptions, none of which say much about Thor's role. The sources he is talking about are the Eddas and Sagas, which were written down some centuries later in Iceland in the 1100s-1300s. This is problematic; things get distorted over time, in particular the prose/oral stories. (poetry is more robust since you couldn't change it without messing up the meter) There's been a strong emphasis on contemporaneity of sources in Viking Age Scandinavian historiography for the past century.

Further, the Eddas are folklore stories about the gods doing stuff. They're almost wholly unconcerned with the details of the actual religion, who people worshipped, how and when and why. You'll find some references to sacrifices (blót); you'll even find the question ("Do you know how to sacrifice?" in Hávamál) but you will not find next to nothing saying much about how they were carried out, much less more minor customs.

But we do have a source saying Thor was a god of thunder and weather, namely Adam of Bremen's chronicle, written around 1070s-1080s, at which time paganism was still around in parts of Scandinavia, particularly Uppsala, which he is concerned with and doesn't consider the Swedish king to have be doing enough about. As Adam described it, there were three idols in the temple there, of Odin, Thor and Fricco (generally read as Freyr), and these were gods of war, thunder/weather and fertility respectively, who were sacrificed to by those who wanted victory against enemies, who wanted good harvests, and those who wanted children, respectively.

Adam of Bremen is however not a primary source on this; he had never seen Uppsala. His account is also unreliable, containing many geographical inaccuracies and some fantastical elements. Yet for lack of much else it's an important source, and well-spread in its own time, it was influential also on those "Norse sources" from later. More importantly, there's not that much to doubt about this particular detail: There's ample evidence from things like place names that those three gods were worshiped around Uppsala (but also for instance Ullr, who's scarcely mentioned in Icelandic sources).

That said, most claims of Norse gods being "the god of so-and-so" are at best built on shaky ground. The god Týr's frequently claimed status as 'god of war' is based almost entirely on him having been assigned the week-day the Romans had assigned to Mars. A competing claim by Dumézil that Týr was a god of justice was based on his own reading of the myth of him losing his hand and because it fit into his conception of a common Indo-European justice-god tradition.

Many are even more dubious than that, for instance claiming Ullr was a god of hunting and skiing, because Snorri's Edda says that he was skilled at those things. More probable (IMO) is that Snorri extrapolated that description of the god from older kennings referring to Ullr as a 'bow-god' and 'ski-god', but that those were originally references to stories about him lost by that point. (e.g. 'one-handed áss' is a kenning for Týr, clearly referencing the story his hand being bitten off by Fenrir. If you imagine that story had been forgotten, perhaps people would put that together with the 'war god' idea and claim he was the god of people who'd lost limbs in war)

So this "god-of-this-and-that" thing may to some extent be a projection of Greco-Roman pantheon onto the Norse one. However, the problem with that, is that Greco-Roman influences on Norse culture go back all the way to the Roman Iron Age (~1-400 AD), named thusly because of the massive influx of Roman artifacts and influences.

At the local level, religious customs among the average Scandinavians were quite heterogeneous, which we know from archaeological evidence, particularly variations in burial customs. The only truly religious practices that were completely "Norse", as in fairly homogeneous across Scandinavia, are those of the elite, who engaged in things like erecting burial mounds and ship burials. It's also doubtless that the sagas (which are largely about the doings of kings and elites) and Skaldic and Eddic poetry were composed by and for members of the elite. So there's every reason to think the elite's religious world-view was overrepresented in the later sources. And this was a religious world that was more shaped by Greco-Roman ideas than that of the common folk.

In other words, the average Scandinavian of the Viking Age may not have been particularly concerned with the whole pantheon to begin with. It's always been pure assumption to believe that what the Icelanders wrote down centuries later was an accurate portrayal of a homogeneous culture, but it's an assumption more challenged today than ever before.

That said, Thor is definitely associated with thunder. How could he not be? He is a personification of it. The name Þórr means "thunder" in Old Norse. Etymologically it's the same word as English 'thunder' too. (Proto-Germanic *þunraz - > þunor -> thunder in English, *þunraz -> þórR -> þórr in Old Norse) To this day, Scandinavian words for thunder reference Thor, e.g. Danish/Norwegian torden ('Thor-noise/din') Swedish tordön and the more euphemistic åska ('travelling god/áss'). Also the Old Norse term reið ('ride') could also mean 'thunder-storm'.

So there's reason to doubt the claims of gods being gods-of-things, but even reason to doubt that the gods were even that important. Yet at the same time, out of the gods, Thor simply cannot escape an association with thunder.

Relations with Norse gods are consistently depicted as quite transactional - if you want something, you should perform a sacrifice (although as Hávamál also says, 'it is better to not ask than to sacrifice too much') Adam of Bremen's account is in-line with this, as is for instance Ibn-Fadlan's eyewitness account of Varangians performing sacrifices. Thor was a very popular god (as seen in for instance place name evidence), and that popularity is more easily explained by him being a god of weather and thus good harvests in these agricultural societies, than by him being a badass who goes around killing giants. Which is entertaining and all but perhaps not reason to (say) give up your best horse to him. (edit: the point here being that the mythological stories about the gods only had the loosest of connections to their cult)

I want to note a couple of other points here.

First, I am not actually sure that "god of X" is a way to categorize Norse mythology. The framework works very well in the context of old Roman religion but it isn't even clear to what extent it works when looking well at Greek myth, certainly not during Hellenic times. But Roman religion was both very ceremonial and functioned without any real mythology. We don't have any stories about Quirinus or Jupiter as Jupiter, or the like. To be sure, the Greek gods in Homer have domains assigned by the Fates -- it was the process of drawing lots which divided the kingdoms of Poseidon, Hades, and Zeus, for example and that process (lachesis) gives name to one of the Fates. But Poseidon being the god of the sea in that respect mostly means he rules the sea just like Hades rules the underworld. The same designation can't really translate into Aphrodite being the Goddess of Love.

I think it is reasonably well established that the Eddas and some parts of some of the Sagas codify something of an oral tradition (though these have been filtered in a number of respects, not least in the very act of writing them down). Part of the difficulty is that oral stories don't really function in this same way. In his field research into oral traditions following WWI in the Balkans, Albert Lord ("Singer of Tales") was able to demonstrate that rather than being repeated, that oral epics were composed anew (and differently) in every performance. What emerges from Lord's work (along with Millman Perry's analysis of Homer) is an understanding that oral traditions are flexible, multifaced, and multiform.

Thorr's name means "Thunder." Of course he has other names associated with noise but none as prominent as that. A second very important point is the relationship of Thorr to grain. Not only do you have the dominant theory that Sif, his wife, is a grain goddess in part due to the myth of Loki cutting her hair only to have it replaced by magical golden hair made by the dwarfs, but more prominantly you have the fact that Thorr's hammer has a name of Mjolnir, which means "grinder" (and is related to the name of the tooth "molar"). Mjolnir can also refer to a grindstone or a mill, thus providing a series of puns etc in relation to the Hrungnir's grindstone during the fight between the two of them.

As far as I can tell, Thorr's original role was as protector and harvest god, with lightening and thunder as attributes in the same way that Zeus is associated with these things but would not be called a god of thunder. Thorr is the Thunderer, but it would be a reduction in his role in myth and probably old Norse religion to *reduce* him to such.