Like you said, it's anec-data. Among my friends, two were very sleepy the following day and otherwise fine. I had a sore arm for two days, and no other effects. My parents didn't have any noticeable effects.
I know this is just another anecdote, but I get a flu shot every year and have never experienced anything more than a sore arm. With 2nd Moderna dose I had a fever of 103 and was completely out of commission for 36 hours.
It's all relative. The side effects from a flu shot are mild compared to the flu. Ditto for the Moderna vaccine.