5. There is no medium or long-term safety data
The Pfizer vaccine was authorised for 12–15-year-olds based on just two months of data.  For 5-11s, it was 2.3 months. 
Medical trials for vaccines normally take several years. 
Time has been saved in the trials by fast-tracking approvals and running certain phases concurrently rather than consecutively, for example.  However, regardless of the resources or funding, there are other elements that are impossible to short-cut.
We only have a few months of safety data. We don’t know if there are adverse side-effects that will emerge after one year, or five years, or even fifty years.
We are still learning about these vaccines. Concerns have been raised about autoimmune disease and possible effects on fertility and the placenta.  A recently published paper raised the possibility that mRNA COVID-19 vaccines could trigger prion-based, neurodegenerative disease. 
An elderly person may not be worried about potential long-term side effects. For children, however, this is a critical consideration. 
Check our references for the above statements:
It’s safer to wait