Cynically, the thing you quote shows no evidence of vaccines actually having any effect and the study author wants more funding.
Less cynically, infections can throw your body's homeostasis totally out of whack over long periods of time. I've had lingering side effects that lasted nearly a year before I was totally back to normal. It absolutely would not surprise me, at all, if covid screws with some people's menstrual cycles. It would surprise me more, but still not incredibly, of a vaccine that provoked an immune response and makes you feel like shit for a day can also throw of your homeostasis for a while. What would surprise me is a f the vaccine did so at a higher rate than covid infection, which is to be clear, not even approached by the linked paper.
All that to say, you've provided an unsubstantiated hypothesis. More than likely, it isn't true. But even if it is, it doesn't contradict my initial statement. Because I didn't say there could be no long term side effects, I said that those side effects would be a strict subset of covid side effects.
Edit: also re-reading, I'm pretty sure she's saying they're be multiple months to make sure the stuff goes away. That is, the side effect, if real, is present and detectable in a month or two.
There's just like really no reason to bribe your going to take the vaccine, have no detectable health concerns for a year and then differently develop something.