I will offer one slight caveat to your last statement, just for complete clarity. A vaccine can cause a long-term side effect. But this is through the damage caused by an event that occurs soon after the vaccine is given. For the example I often give of Guillan-Barre syndrome, although those that survive usually mostly recover, they will often suffer from long-lasting side effects that are the consequence of lying paralyzed in a hospital bed for days to weeks. If a vaccine triggered the syndrome, the onset happens relatively quickly, but the damage can endure.
I know, semantics, but an important distinction when we're dealing with malignant disinformation attempts.
Your point stands, though, that we simply have never observed problems with vaccines that present themselves unannounced months to years later.