Female warriors

The discussion about the Birka warrior-woman's grave is largely misconstrued, especially online and in lay discussions who conflate warrior as in 'one fighting battles' with warrior as a social status. (no small part of the uproar was plain mysogyny though, quite evident in the hate-mail Charlotte Hedenstierna-Jonson got)

I mean, if you found a modern grave with a person that had a military uniform and regalia, you'd assume it was a soldier's grave. But that person's status as a solider (in a modern context) comes from serving in the armed forces. It doesn't necessarily mean the person ever saw combat. So even today, warrior status doesn't necessarily mean you've been to war.

It is not so different here. I'd say its fairly well established that warrior status did not necessarily have to do with whether one had fought or even was capable of doing so well. There are (royal) weapon graves for children. There are weapon graves for the physically handicapped. These people nevertheless had warrior status, probably due to their social rank and family.

The conclusion of Hedenstierna‐Jonson et al was not that this warrior grave proved Shield Maidens were a thing and that women fought in battle, nor that this one woman did so. Rather the implication is that women, too, could have warrior status. Which would likely due to their birth, not their prowess on the battlefield. Notably she was also buried with other high-status items, such as a chessboard and several sacrificed horses. It's not really that controversial and a fairly straightforward interpretation of the find. (where they get a bit more speculative is suggesting the chessboard might've implied she had reknown for strategic military thinking)

This isn't that much at odds with what was already known either; there are a number of references in literature and also early medieval law codes that imply women (of higher social ranks at least) could run their dead husbands' estates and manage the property. We have a small number of runestones erected by women after husbands and sons, rather than by the sons and brothers of the deceased.

Vikings

Let's be clear: There was no "Viking" people. According to written records, Danes were Danes, they called themselves Danes. Norwegians called themselves Norwegians. Scandinavia was not more homogenized back then but quite the opposite, there were stronger regional identities (Jutes, Gotlanders, Östgötar and Västgötar, etc) Perhaps even to the extent of being more important than the aforementioned 'national' identities. (see below)

Old Norse "víkingr" was the term for pirates, raiders, traders, those who went off on journeys. Viking Age inscriptions indicate the macho man would be more likely to call himself a drengr (literally 'man', more broadly a 'badass') than a viking. Mats G Larsson has even argued it would've been an insult to most to call them a viking. Out of thousands of rune stones from the Viking Age, only five (DR330, DR334, G370, Sm10, Vg61) mention someone being a viking or going on a viking journey. One (U617) mentions someone being on a defensive watch against foreign vikings (many references to those exist in later Saga literature; even black ones). 16 inscriptions mention Vikingr as a personal name. (that could be evidence the term wasn't offensive, but that's complicated by the fact the name could've had a different meaning/etymology)

If you're wondering how such an obscure term could become elevated to the status of demonym, and how everyone (outside Scandinavia) learned about vikings in the first place: It's from 19th century romanticism when the Victorians more or less invented the Vikings in their modern popular conception (from horned helmets to burning ship funerals) and projected their own values onto them, from the 'noble savage' to anti-catholicism. (see Wawn's definitive book on that topic) Also the Scandinavians themselves had a big part of constructing their own Viking mythos in the 19th century, although it's primarily the former to which the international reknown is due. And it's the romanticized and largely fictional image that people are interested in, which is why we get questions about 'vikings' here and not about what 11th century Bornholmers were doing.

What does that have to do with burials?

Everything. Just as "viking" doesn't imply an ethnicity, it doesn't imply a religion either. Plenty of vikings where Christian, and increasingly so in the second half of the Viking Age. Out of the aforementioned five runestones for vikings, two are explicitly Christian (G370, Sm10), and none obviously pagan. (in terms of dating they could go either way)

But, what I'm working up to here is this: You cannot generalize about just the 'pagan' burials either. Because was no single pre-Christian Norse religion with a single set of burial customs. The very notion that such a thing existed is the result of the aforementioned 19th century and its agendas.

This was argued specifically on the basis of burial customs by Fredrik Svanberg in his quite important and influential two-volume PhD thesis. Svanberg's point is that there was no common Scandinavian culture of the Viking Age. His extensive survey of burial practices demonstrates the strong regional variations, such as over a half-dozen burial practices in Scania alone. He argues not only against a common Scandinavian culture but (as said above) in favor of small regional identities and against even the concepts of Danes and Swedes.

The latter point is perhaps going too far, and the fact that a national supra-identity is not at odds with a stronger local one has been argued. (e.g McLeod, Sindbæk)

In any case, the opinion of most historians these days on what Viking burial practices were, the answer would be that the very notion of a single 'viking burial practice' is an artificial construct. Not everyone is on-board with Svanberg's interpretation of that as making a mockery of the entire concept of the Viking Age and pan-Scandinavian identity, but the point on burial practices nevertheless stands. For instance, Neil Price is one who thinks Svanberg goes a bit far, but nevertheless states (2007, page 257):

Perhaps the central element of Viking Age Scandinavian funerary ritual was its individual character. After more than a century of excavations there can remain no doubt whatever that we cannot speak of a standard orthodoxy of burial practice common to the whole Norse world.

In short: People were buried with different things in different places. For instance in Öland and only there, there was a custom of putting fossils such as ammonites (mollusks not middle-easterners) in graves.

Nor can one, as that pretty bad History.com article does, generalize women in 'viking society'. While certain women may have achieved high status and been allowed into otherwise male-only spheres, that should not distract from the fact that the same societies also held other women as slaves that were raped, abused and (if Ibn Fadlan is to be believed) murdered as human sacrifices, at the whims of their owners.

But it's all part of the old Victorian tradition of projecting contemporary values onto the 'vikings' they'd made for themselves.

References:

Fredrik Svanberg, Decolonizing The Viking Age I and Decolonizing The Viking Age II: Death Rituals in Southeast Scandinavia AD 800-1000, Acta Archaelogica Lundensia, 2003

Shane McLeod, Know Thine Enemy: Scandinavian Identity in the Viking Age, 2007

Søren M. Sindbæk, The Lands of Denemearce: Cultural Differences and Social Networks of the Viking Age in South Scandinavia, Viking and Medieval Scandinavia, 169-208, vol 4, 2008

Neil Price, _Dying and the Dead: Viking Age Mortuary Behaviour’, in N. Price and S. Brink (eds), The Viking World, London, 2008, pp. 257-273

Charlotte Hedenstierna‐Jonson, A female Viking warrior confirmed by genomics, American Journal of Physical Anthropology, v164, Issue 4, 2017

Andrew Wawn, The Vikings and the Victorians:Inventing the Old North in Nineteenth-Century Britain, Boydell & Brewer Ltd, 2002