Science versus conspiracies: thank you so much! My 66-year-old wife got her Pfizer booster and flu shot on the same day without side effects other than a slightly sore arm. I received mine at different intervals (age 70). But neither of us thought about this possible consequence.
What is important to remember is that even if a connection is demonstrated (which as you say hasn't been), it is also much more important to remember the millions who died because they couldn't or wouldn't get the vaccine. And get it!