Argument from needless suffering

1. God is all-powerful, all-knowing and all-good.
2. If God exists, people wouldn't suffer needlessly from disease, accidents and natural disasters.
3. People suffer needlessly from disease, accidents and natural disasters.
4. Therefore, God does not exist.

The only way to refute this argument is by attacking the 2nd premise. I'm going to reply to what I think might be the main objections to it:

• People are being punished by God
• It's trivial to find examples of children suffering needlessly to an extent that, if it results from God's punishment, would only prove there to be an evil God.
  
• Suffering exists for a greater good. This might be split in two:
• If the objection is that God inflicts pain to a person so that he can feel happiness later, I could as well say that happiness exists so that a greater evil can occur, the higher you rise the harder the fall. There doesn't seem to be any consistent pattern that supports this type of direct intervention and denies the other so this doesn't relate at all to our observation of the real world.
  
• If the objection is that some people have to suffer so that others can be happy, then that's just not consistent with the 1st premise. An all-powerful being would certainly be able to avoid the suffering of some without producing more suffering on others.
• God created humans with free will and does not interfere in our free will at all.
• This is what I consider the most persuasive of these objections and this is what I've seen being used but note that I've assembled the 2nd premise precisely excluding human interaction. An earthquake is not the product of free will and a god that doesn't avoid the kind of suffering that it provokes needs to be different than the one described in the 1st premise. If the first premise is denied then we wouldn't be talking about the usual concept of God from the main western religions.
• In order for happiness to exist, suffering needs to exist, otherwise happiness is meaningless.
• How would heaven be consistent with that principle then? If there can't be happiness without suffering and if there is no suffering in heaven, then there is no happiness in heaven.