That's not true at all. If you (like me) live or work in the middle of NYC, for example, and an incoming Russian ICBM comes, you have an almost 100% chance of surviving, if you act.
To see this, let's use the nukemap [1]. I modeled a Russian Topol nuke of 800 kT. Most of the actual warheads on Russian nukes are between 550 and 750 kT per wikipedia, but let's use 800 kT (because it's in the dropdown menu). Ground zero is lower Manhattan (City Hall). Nukemap estimates the number of fatalities for an optimal airburst explosion to be 1.44 million people. And that number might be accurate. But it reflects the assumption that everyone just does nothing to try to stay alive.
But you can stay alive with this very simple trick: when you hear the smartphones blaring the nuclear alert pings (which will probably sound quite similar to the Amber alerts), head to a subway station. Bonus points if you pick up a bottle of water. Extra bonus if you happen to have a Covid facemask in a pocket. Extra-extra bonus if that mask is a KN-95 or similar, but even a simple surgical mask is ok.
But won't an 800 kT nuke destroy the subway system? And even if it doesn't, won't the radiation kill you after that?
The answer to both questions is no.
The fireball radius of the nuke is (according to Nukemap) 0.88 km. But the optimal height for the airburst is 2.9 km, so the fireball won't touch the ground. The subway ceilings just below the detonation might cave in, but those a little bit away from it will probably not, since NYC has so many tall buildings, which would absorb a lot of the blast energy and shield the ground.
The other effects listed by Nukemap are moderate and light blast damage and thermal radiation damage. None will affect you if you are underground.
How about the radioactivity and the fallout? The radioactivity after a nuclear bomb explosion decays very quickly. Here's a quote from the website of the city of Hiroshima [2]:
Residual radiation was emitted later. Roughly 80% of all residual radiation was emitted within 24 hours. Research has indicated that 24 hours after the bombing the quantity of residual radiation a person would receive at the hypocenter would be 1/1000th of the quantity received immediately following the explosion. A week later, it would be 1/1,000,000th. Thus, residual radiation declined rapidly.
But what are you going to drink and eat during that week?
Humans don't die of starvation in one week. But you need to drink water. You may be able to find some liquids in vending machines, but they won't be enough for everybody. As I said, it would be great if you grabbed a bottle of water when you heard the alarm. But you might need to share it with the other people in the subway, so in the end water will be a problem.
Here's what you do: stay put for 24 hours, preferably 48. People don't die of thirst in 2 days. Hopefully some people (fireworkers, National Guard, FEMA, etc) will come and bring water. Of course, those people will have lots of issues on their hands, and even if they come, they might not come exactly to your place. If no help comes, be prepared to dash outside after 24 of 48 hours, and look for water. If you stay outside for just one hour or 2, you might get some type of radiation sickness, but you won't die. Use a facemask. The better the seal, the better your chances of avoiding cancer in the future. Outside you might find fireworkers and other people busy looking for survivors, they might help. If not, head for the river. Get water and get back underground, not necessarily in the same subway station.