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The answer that seems to be emerging from several different lines of research is that a) they always had fairly low fertility and b) they didn't really go extinct as such, they just intermixed with Homo Sapiens Sapiens and because the later had much higher fertility, Neanderthal genes got diluted down to the present ~2% in the Eurasian population.
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I thought even after the merge the Neanderthal genes continued to get rarer, indicating natural selection against them
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If it's 2% now after 2000-3000 generations, it must have stabilized because any number <1 is basically zero when raised to the 2000th power.
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Sounds plausible indeed. Anyways, neanderthals operating a large scale fat production 125 thousand years ago could be a good plot for another hollywood movie scenario. Any takers?
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You might enjoy Hominids by Robert Sawyer
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I thought it was mostly because our ancestors murdered them?
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Common misconception, more likely outcompeted
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Doesn't outcompete include murder? We are a very tribal species, and the history is full of genocides and mass murders, so from a very uneducated viewpoint, this sounds reasonable.

If not that, is it that we depleted the resources they depended on?

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Not necessarily, it could also mean that homo sapiens was just more successful - better fed, bigger population, etc. It's not likely that early sapiens was so organized that they intentionally genocided neanderthals, it's more like they were subsumed etc. A slow process across thousands of years.
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Outcompeting includes murder, rape, war, and cannibalism. But we have population overlap for millennia, so that’s kinda factored into numbers.

All primates are resource competing, so outcompeting is also drinking up their milkshakes. But, again, that’s the baseline.

Non-conclusively, from my lay understanding, the tail end of falls into general bi-lateral competitive practices and breeding rates leading to ‘us’ not ‘them’. All columns all the time, not one crisp incident or behavioural change.

[And there’s no indication that ‘they’ geno-rapo-ate us any less than we them… if being slightly better at mass murder was the difference, then yay for our side?]

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Great question. When people say outcompete it can certainly include violence but we’re talking about populations spread over continents over thousands of years. Factors like technology, fertility, adaptability, etc. are more what people mean when they said outcompete.
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You really think we would have let a competing species exist?
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Depends on whether they were considered competing, and whether "we" were as organized, single-minded and competitive as this statement seems to imply - "we" probably weren't, not until larger kingdoms and empires started forming ~4000 years ago.
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Lions, bears, wolves, etc all survived us
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Barely and only because some of use decided to protect them.
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Bears and wolves were indeed "removed" from parts of Europe by humans.
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