The evidence that showing vaccines are safe and that they save lives is generally overwhelming, so I'm always pleased to see another article reviewing the data behind them. I figure such articles will lead to even more people being vaccinated and more lives saved.
However, I was disappointed that this recent New York Times article did the statistics so poorly. The article compares 10,000 people who got various diseases with 10,000 people who were vaccinated. This comparison is inappropriate, because most people who do not get vaccinated do not get the disease they are being vaccinated for, and, especially for diseases like the flu, many people who do get vaccinated get the disease they were vaccinated for. A proper comparison would compare some number of people who were vaccinated against the same number who were not vaccinated.
I use the CDC figures, and the figures presented in the NY Times article to do just that, focusing on the flu, since it is by far the most common illness mentioned in the article. Millions of people in the US get the flu every year and typically tens of thousands die from it.
It is known that the flu vaccine reduces the chances of getting the flu by about 50% (see https://www.cdc.gov/flu/vaccines-work/vaccineeffect.htm). The percentage of the people who get the flu varies quite a bit. In 2017-2018, an estimated 45 million people got the flu (more than 10% of the population) but in 2011-2012, only about 9 million people got the flu (see https://www.cdc.gov/flu/about/burden/index.html, which also has hospitalization and death rates). So, we'll assume a year that is somewhere between the best and worst years, where about 1 in 13 people get the flu. Given a vaccination rate of 50% (sadly it has been below this recently), that means that, you have about a 1 in 20 chance of getting the flu if you get the vaccine and a 1 in 10 chance if you do not.
Now let's consider the effect of everyone in the US (about 300 million people) not getting vaccinated versus getting vaccinated. The following table summarizes the results (using the per 10,000 people figures found in the NY Times article but projecting them to the full population).
This shows that vaccinating everyone would mean fewer than 270,000 hospitalizations (versus 540,000 if no one was vaccinated) and fewer than 21,000 deaths (versus 42,000). The only area where the vaccine is worse is "other bad effects," where I am grouping allergic reactions and Guillain-Barré Syndrome. In this case, about 800 more people might suffer these (sometimes) very serious side effects. However, this pales in comparison to the more than 20,000 lives saved by the vaccine annually.
Even these huge benefits are likely understated. If everyone were to get the flu vaccine, it is likely that it would spread less, since many of the people who currently catch the flu who were vaccinated get the flu from people who are unvaccinated (in 2017-2018 only about 37% of adults were vaccinated: https://www.cdc.gov/flu/fluvaxview/coverage-1718estimates.htm). Also, vaccinations have been shown to reduce the severity of the flu for those who get it (see here again).
So overall, despite the poor comparisons in the article, which seems to imply no one would die from the flu if vaccinated, the benefit of the flu vaccine is still overwhelming.