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Yes, vaccines — as with all medical procedures — have bad outcomes for some people. You can't inject everyone with saline without someone having a bad reaction, if only because it ended up stabbing a tendon or something.
However, the risk of injuries for all common vaccines are orders of magnitude less than the harm from the things they decrease. For ever 1 person harmed by a vaccine, 100 people don't get seriously ill from the disease it counters. And while it’s an enormous bummer for that 1 person, the other 100 experienced a modern miracle of public health.
No one’s ever said vaccines can't make you sick. (Well, they've said that they provably do not cause autism, but that's a different claim.) They've said that if you go to Vegas and have an option to bet where you have a 99% chance of winning $1000 and a 1% chance of losing $1000, you take that bet every time.
Except all the people who ran around yelling "100% safe and effective" four years ago.
Perhaps some non-experts were spouting such nonsense, but nobody should be putting any real importance on what they say.
Tap water is safe, but non-zero people get sick from it. That just means it's not completely safe.
Penicillin is effective, but some germs are resistant now. That just means it's not completely effective.
Public messaging around human health, especially with respect to powerful tools that modify the immune system, has always been extremely challenging. Public health officials typically want to maximize health for a whole population, and we know that messaging can lead to society-level changes in behavior around vaccination. So experts are typically quite careful when they talk about harms from vaccines.