As of mid April of the 75,000,000 people who had been vaccinated, 5,800 had gotten Covid, and 74 had died as per the CDC. That is dying by Covid. I have not seen data for deaths attributed to the vaccine itself, but if we are now at 150,000,000+ having been at least partially vaccinated and 340 have died from that, well, that's pretty low. I can't find reliable data, so I will trust your numbers.
On the other hand 1.5 million children (edit... as of early May 3.6 million) have gotten Covid, and 340 have died as per your numbers Edit - WSJ reports 279 deaths, so even lower. (I will trust them). Very low chance of kids dying from the 'vid, but probably much lower chances of them dying from the vaccine. It is obviously way too early to tell if there will be any long lasting affects from vaccinating children with a new vaccine.
The physicians and microbiologists seem to feel the odds are very low... they could be wrong.