Yes, the rate of hospitalization would be a more useful measure than the rate of infection.The real statistic would be hospitalizations.
However, as with infection rates, the question we need to answer to measure the effectiveness of the vaccine would be:
What percentage of those fully vaccinated were subsequently hospitalized ?
Knowing what percentage of those hospitalized had been vaccinated tells us little or nothing, and only misleads the simple-minded and credulous like @Clarence Worley .
To illustrate further, imagine if someone told @Clarence Worley that drinking milk as a child caused brain cancer later in life.
They might quote a statistic that 95 % of all adult brain cancer patients had drank milk when they were children. A simple fellow like @Clarence Worley might take that as overwhelming evidence that drinking milk as a child causes brain cancer as an adult.
In actual fact, the question that would need answering to check this hypothesis is: What percentage of children who drank milk as a child went on to develop brain cancer as an adult.
Simple thinkers like Clarence (if we can even call them thinkers), often fall for this backwards application of statistics when it comes to cause and effect. This leads them to ramble on about the percentage of infections (or hospitalizations, or deaths) that were vaccinated, as if this was a meaningful statistic.