How is that bad? A vaccine is designed to cause a response by the immune system; this is certainly not pleasant, but exactly the trade against the potentially much worse effects of an actual infection.
Feeling “slightly under the weather for a couple of days, similar severity to having a cold” is what Covid looks like in most children.
Covid infection is “potentially” much worse, that’s true, but so is a drive to a vaccination site — the child might die or get seriously injured in a traffic accident. What matters is not the “potential” but rather the actual likelihood, the expected value. Has Covid been dangerous to children? The answer is, rather overwhelmingly, no.
