At least for beaver, free and ecological illiteracy does paint most of the picture (though, in fairness, ‘extermination’ may be too strong)
Do you have any references about this? It's the first time I heard of it and googling for "the genocidal starvation program" only gets me hits on that other genocidal starvation program, the Holodomor.
See also https://en.wikipedia.org/wiki/Bison_hunting#19th_century_bis...
Specifically:
The US army and settlers moving West executed buffalo because Indigenous peoples in the Great Plains depended on them. 'Every dead buffalo is a dead Indian' is one of the quotes from the time. The army/settlers would also poison the buffalo carcasses so Indigenous people couldn't eat from the dead carcass, and as a consequence, other animals in the environment (e.g. wolves) couldn't eat the meat either and likewise suffered, and further reduced the food supply.
This took place after the 1868 treaty of the Black Hills. So the US signed a treaty acknowledging Indigenous land and sovereignty. And then started a genocidal campaign, against the people they just signed the treaty with -- tresspassing on treaty defined land to do so and kill the buffalo. The Supreme Court acknolwedged the history of these wrongdoings in a ruling in 1980, awarding financial compensation to the Indigenous peoples -- however the Indigenous plaintiffs did not want monetary awards, they sought rematriation of their stolen land and the upholding of their rights under the 1868 treaty. The battles to uphold those treaty rights continue to this day.
(In 2007, the UN ratified the Declaration of the Rights of Indigenous Peoples, which affirms Indigeous claim to treaty rights such as the 1868 Black Hills treaty (albeit in a 'non-binding' way). The declaration has now been ratified by all UN member nations -- US, Canada, Australia, and New Zealand originally held out for several years -- and many groups are now operating with that document as a legal/political framework and justification to take their treaty and Indigenous rights cases forward.)
Now, if the Inuit had tried to get extremely wealthy by using their rifles to hunt seals and caribou as the basis of a large and ever-growing export industry, they'd have indeed destabilized the ecosystem and there would have been a collapse in the population of seals and caribou, as seen with European hunters of wildlife in North America with beavers and bison and passenger pigeons. If that had happened, the Inuit themselves would most likely have followed the local animals into extinction.
[edit] Note with bison, there was a deliberate extermination policy pursued by the US government as a means of destroying the food supply of various indigenous tribes that relied on bison for sustenance (aka the genocidal starvation program). This doesn't explain the extermination of beaver and other wildlife populations, however.