How did Neanderthals survive so long?

Anthropologist and Neanderthal-defender John Hawks write about how little is known about Neanderthals, and adds:

Across the entire timespan of existence of Neandertals and the branch that gave rise to them, probably fewer than 50,000 of them existed at any time. I would not be much surprised if the true number was much smaller. If the average lifespan of a Neandertal was 20 years, maintaining a population of 50,000 individuals would require around 7 births per day. For the more than half million years this population and its ancestors existed, back past Sima de los Huesos to their common ancestors with Denisovans and African peoples, we can say there were as many as 1.3 billion Neandertals.

Anthropologists speculate endlessly about why the Neanderthals went extinct, with the leading theory being that they were raped and murdered by African migrant hominids who were invading Europe. The only survivors were Neanderthal-African hybrids.

The previous theory was the Out of Africa theory, but that has been disproved by DNA evidence.

I think the bigger mystery is how the Neanderthals lasted so long. The lived for 0.5M years with small populations in Europe where regular ice ages made it nearly uninhabitable.

A Wash. Post article says that polar bears could be wiped out in a manner analogous to Neanderthals:

Many humans carry traces of DNA from Neanderthals, which means we’re all hybrids. ...

Amstrup has studied bears in the Arctic since the 1970s and was instrumental in helping list the polar bear as a threatened species in 2008. He, like other experts, characterizes this “new” bear relationship as more beneficial to grizzlies than polar bears. That’s because there are more grizzlies than polar bears and because grizzly territory is expanding while polar bear territory is contracting. What that adds up to is a good chance grizzlies could essentially dilute the polar bear population until it doesn’t exist at all, they say.

Polar bears are getting the short end of the stick in this relationship, not “gaining any genetic diversity,” said Geoff York, who led research on polar bears at the World Wildlife Fund for almost a decade before joining Amstrup at PBI. ...

All hybrids that have been analyzed had grizzly fathers, because grizzly males roam to establish territory and come in contact with receptive female polar bears. Female grizzlies tend not to stray far from their home ranges, and male polar bears don’t usually creep into grizzly habitats.

The Neanderthals did not know what they were up against. Neither do the polar bears.

If this were happening to a modern human society today, what would it do?