I would disagree with your initial statement "Many cultural cuisines like rabbits, cows, trout, pigs, chicken, reindeer or walruses are consumed without international criticism", at least in spirit.
There are protests, within and across national boarders, about the hunting or eating of certain groups of animals, including those in your list. Cows are farmed, not hunted, but several groups protest against this (such as PETA). There are also multiple reports of people being killed by lynch mobs in India on suspicion of harming or slaughtering cows, for example.
There are also protests and laws protect various endangered animals from hunting, including seals(which several populations claim hunting of as part of their cultural heritage), trout(highly regulated where I am from), and rabbits, such as the Amami rabbit (Pentalagus furnessi) of Kagoshima Prefecture of Japan.
Instead, I would say that whale hunting is the "loudest" protest against hunting, especially on the international stage for a couple of reasons:
1) Whales are endangered, and are actively hunted. I think, although the resources collected are different, a good comparison might be the hunting of ivory of elephants and rhinos, which garners a large amount of international press and condemnation, at least where I am.
2) Whales are "photogenic", and thus garner more public support: better adjectives might include cool, majestic, exotic or cute. This is a general trend: we hear more about efforts to save, e.g. endangered Pandas, turtles, elephants and gorillas, then we do about efforts to save equally or more endangered species of mice, ferrets, crustaceans, spiders(kill them with fire!), or flies.
3) Whales live in/travel through international waters. As such, lobbies for their protection need to be international in nature to be effective. Lobbying for the protection of animals that have smaller geographical footprints tends to be more effective if targeted to a specific nation or nations. For example, monarch butterflies migrate between the US and Mexico, and so there has been a lot of successful lobbying to grant them protections on both sides of the border.