The samples you gave make no sense.
Firstly, a person suffering from ADHD can do the dishes, despite the odds being against them. And a paralyzed person can raise their hand, given advances in technology.
And these are bad examples, as being incapacitated isn't an argument against free-will. The paralyzed still wants to raise their hand, even if they are unable at the moment. And the far more difficult question is if that want was entirely predetermined by the universe or not.
Quantum randomness and the huge number of variables at play point to the fact that believing that we don't have free-will is a non-falsifiable notion.
Consider this thought experiment: God comes to you and says ... "Here are 2 universes you can observe, one in which people have free-will, and another, that looks similar, but its people are just sophisticated automatons. Make an experiment to say which is which."
Your irrefutable argument quickly falls apart because there's no way you can show any relevant evidence for it. And it's worse than talking about the absence of God, because we experience free-will everyday. It's like talking of consciousness — we can't define it yet, but we know it's real, as we're thinking and talking about it. And a theory being non-falsifiable does not make it false. God could exist and you could have a soul.
You may or may not believe in free-will. But not believing in free-will is dangerous because it makes one believe that everything is predestined, so there's no point to doing anything, no point in struggling to achieve anything, no point in trying to escape your condition. If no free-will is possible, does life have any value at all? And that's the actual philosophical bullshit.