The article in that link to the conspiracy site you posted is published by a mechanic from England who doesn't really seem to understand which denominators matter. If you see something like x% of the sick/hospitalized/deceased were vaccinated, the better the vaccine uptake the scarier this number will seem! It is using the wrong denominator.
For example, in a scenario with 90% vaccination rate, 4 people get sick: 2 vaccinated 2 unvaccinated: In this scenario, 50% of the sick were vaccinated, but this is the wrong metric to look at! It is using the wrong denominator. It doesn’t take into account that 90% of the population is vaccinated. This is called flipping the conditional (and is a problem that we see all the time!). Instead of looking at the probability of being vaccinated given you are sick, you want to look at the probability of being sick given you are vaccinated.