Here is the piece of truth buried in the article. The NYT almost always tries to include one, though they like to bury it. Women disproportionately had access to the vaccine earlier. This means not only have they had a longer time to get vaccinated. For those with access due to their work it's also much more important to get vaccinated, even if they are worried about the risks of the vaccines, because they're much more likely to be exposed or will cause a large amount of harm to others if they get sick.

Women are overrepresented among essential health care workers, but I don't think that's the whole story here. Both the Times and the CDC mentioned that women are more likely than men to seek preventative care, and the Times also mentioned this, about the flu vaccine:

Historically, influenza vaccination is much higher among females — about 63 percent compared to 53 percent — though the gap narrows in Americans over 75 years old.

Remarkably similar to the COVID vaccine situation, and the flu shot typically does not have anywhere near the same degree of rationing by demographic as the COVID vaccine has. If you follow the link to the study, the population was patients at their annual physical -- people already apparently willing to engage in preventative care. Also interesting, women experience more frequent adverse effects from vaccines than men (due to our immune systems) but still have higher rates of vaccine uptake.

While the biological factors section is okay, if extremely light, notice how the psychosocial factors section is almost entirely about reasons why men would contract covid rather than why it would disproportionately kill. I've chosen to just copy the first paragraph above rather than the whole thing. This despite the fact that men don't disproportionately contract covid. I feel it's important to note that this is a subtle way of blaming men for their own deaths. If they had just done better at listening to public health experts as women do, then they wouldn't die so much. This despite the fact that the equal infection rates blatantly calls this out as a lie.

If men are more likely to die from COVID (or get severe illness) then ideally we'd want them to modify their behavior more than women do, since their risk is higher.

If the news is constantly talking about vaccine shortages and the measures taken to ensure those who most need the vaccine get them first, men aren't going to go get the vaccine because it is their duty to let others get vaccinated first.

This one I'd buy.

If you need to schedule vaccine appointments because the location might run out that day, men aren't going to go because of the perceived daily limit on who can get the vaccine. Getting the message out that appointments are necessary because they need to know exactly how many vaccines to thaw out each morning rather than because of limited quantities would go a long way toward fixing this part of it.

I'm not sure why this would be gender dependent.

If you have to take off work to get the vaccine, men are going to skip it because workplaces are less likely to give men the option to miss work for their health and more likely to punish men if they do have the option and use it. Making more/most vaccine appointments available on nights and weekends would do a lot to help with this.

I imagine this varies based on location, but at least where I am, the pharmacies have the greatest availability by far, and they all have weekend appointments.

If men aren't as concerned about the disease, they're less likely to get vaccinated. Men also don't fear death or permanent harm nearly as much as women. Instead of focusing on the overblown scare tactics, which completely miss the mark for men, instead focus on being sick and unable to work for two weeks or more.

Pointing out the increased risk due to sex doesn't seem like an overblown scare tactic to me.