So off the bat, I'm a little skeptical that the Great Auk was particularly tasty - ie, that it was hunted to extinction for its deliciousness. Anyway this will be kind of a science-y answer but I'll try to ground it in some environmental history.

Great Auks, although flightless and being the original "penguin" (it's a Welsh name for the bird), are in fact (as the common name states) a type of auk, which is a broad family of sea birds including guillemots, murres, and probably most famously puffins. These birds are widespread in the Atlantic and Pacific, and are still routinely harvested - although numbers are declining and hunting restrictions have been added, even in the 1990s there were hundreds of thousands of murres being harvested in the Canadian Maritimes and similar numbers of puffins harvested in Iceland. You can still eat Icelandic delicacies such as smoked puffin or raw puffin heart today in the country. By the way, the razorbill is considered to be the Great Auk's closest living relative. It's smaller and can fly (the Great Auk was the only flightless member of the auk family), but it's superficially very similar.

OK, so it probably didn't really taste much different than its relatives - puffin for example is supposed to be livery, slightly fishy, and/or like beef jerky. Of course people clearly did eat them on both sides of the Atlantic in prehistoric and historic times. But what then happened in the last 3 centuries or so of the species' existence when its numbers crashed to extinction?

A number of factors. First there should be noted some details about the species. It was flightless, and mostly lived at sea, coming to specific nesting grounds in order to breed. These areas needed to be free of predators, and notably climactic changes in sea ice meant that different breeding colonies were at risk from polar bear predation. It also had very particular requirements for which islands fit the bill - they tended to be extremely rocky with particular sloped shores, so it was always a very limited number of geographic locations, which by the 18th century were reduced to Funk Island off of Newfoundland, St. Kilda off of Scotland, and Eldey Island off of Iceland, and a few smaller colonies in the nearby vicinity of these places. So even when the numbers were stable (and premodern estimates put the species population around a couple million), they still were very particular in their nesting sites. They weren't just everywhere.

Next, it's true that in the modern period they were hunted for meat. But notice where the nesting grounds are - they are near areas that historically have been heavily fished. So in the modern period it wasn't necessarily that people were hunting large numbers of auks to sell their meat, let alone for its supposed taste, but they were hunted because it was fresh: ie, ships would stop at these islands and provision themselves with easily caught birds, sometimes by just literally rounding them up and herding them to the ship (this is at least how the stories go). These ships themselves would be bringing salted fish back as their catch, so this was more for their personal consumption.

Of course auks did have more transportable products of value, especially their feathers, which were used for down (much like eider down, eiders being a sea duck of the North Atlantic). Often down harvesters would apparently just pluck the birds and not bother to kill them, but leave them to the elements. Other times the birds, with a high oil content, were actually used as a source of fuel rather than food. However they were perhaps most infamously pursued for their eggs. Here again, demand did not need to be particularly high to massively impact auk numbers - auks laid one egg a season, so just a heavy collection season on the few nesting sites could potentially wipe out an entire year's generation of potential chicks. Interestingly, it was the auk's increasing rarity that made the egg trade worse: by the late 18th century it was getting obvious that the auks were getting rarer, and it made collected egg specimens more valuable to collectors (and hence to hunters). While there were laws on the books in the 18th century in England and Canada against hunting great auks, they clearly did not have great enforcement mechanisms, and there were large loopholes - you couldn't hunt them for feathers or eggs, but you could still legally hunt them for other purposes.

So: yes, people have eaten Great Auks back into prehistory (archaeological evidence shows them roasted at campfires, or puddings made from the eggs). But the big hunting of the last three centuries' of its existence would have seen them eaten by summering sailors on or off their ships, more likely eaten smoked or even raw. And even then it was much more actively hunted for its feathers, for use as fishing bait, as a fuel source, and for its eggs, both as also a food source and as a valuable collectible item. Furthermore, even with a population in the millions before its crash and extinction, annual hunting rates didn't need to be super high to have a devastating cumulative effect. A scientific study by Thomas, et al, 2019 estimated that an annual hunting rate of over 10% of the total population (or some 200,000 birds annually) would have been enough to drive it to extinction. This is high, but not crazy high, and is in range with both the number and total population percentage of similar seabirds hunted in the North Atlantic to the present (and keep in mind those others fly, and have more and more widespread nesting sites).