IMO, if they're going to make a push for it to be an annual thing, it would be a good idea if the shot were available before school starts in the fall.
This is true, but we also know that kids are a disease vector that especially kicks in after every break, and that adults seem to be more vulnerable than kids.
Messaging on this would suck, but I'd love to see split vaccinations, with kids getting vaccinated in late August. That way, they're less likely to vector the disease and we might tamp down the winter surge, and their 'prime' resistance would run through Christmas. And adults getting vaccinated at the end of October, with their 'prime' resistance running from just before Thanksgiving through the end of the winter heating season.