You aren't doing any better job of communicating than the people who you are trying to correct and the message you are spreading is more easily used to prop up

You aren't doing any better job of communicating than the people who you are trying to correct and the message you are spreading is more easily used to prop up harmful misinformation. I have seen very few if any claims that say "vaccinations reduce your chances of death if you are hospitalized." I have provided a citation showing that vaccinations reduce your chances of hospitalization if infected. This is also easy to infer from the wide number of studies that show higher (per time) rates of protection against hospitalization than against infection. The effects may disappear in any numbers (such as those discussed in the article you linked) from places / times were vaccine availability is limited and resticted to vulnerable groups. This is due to a combination of the selection bias and the small numbers of infections among the vaccinated.