They, if you studied, you would know that it wasn't that God doesn't answer prayers. You look at the results of that paralyzed person and assume that it was God who did not answer and yet you don't even know all the nuances to the situation. You don't know their heart, you don't know what they are thinking.
What we do know is that Jesus DID want to heal in his hometown but couldn't but you want better results than Him?
So, if that is what you learned, you learned something that John would be in agreement with.
1 John 5:14 And this is the confidence that we have in him, that, if we ask any thing according to his will, he heareth us:15 And if we know that he hear us, whatsoever we ask, we know that we have the petitions that we desired of him.
So, I have learned that Romans 3: What then? If some did not believe, their unbelief will not nullify the faithfulness of God, will it? 4 May it never be! Rather, let God be found true, though every man be found a liar, as it is written, NASB
I have no idea about the mother as neither do you. We don't know how she prayed, what she believed, was she taught wrong, why the child died and went to a better place, etc. But we do know that Jesus bore our pains and carried our sicknesses. We do know that God wants all to be saved, did all He could to provide salvation to everyone yet people still don't want it and deny what He did. God's fault? I don't think so.
You create your own destiny. Yes faith is one of many components. but as you noted... it wasn't God that failed to do miracles, it was man.
So God does answer prayers... man messes it up.
Well, that paralyzed person is an orphan child and was hit by a car when she was around 16 years old, now she is more than 20 and she is still paralyzed, as if life was not hard enough for her, she also had to end up being paralyzed, there's was no wrong from her part, she was just a girl, on a rainy day going to school.
As I said, faith is the main ingredient of prayers being answered! You know, otherwise, Jesus, the Christ, would have not said this - Your faith has saved you!
Ohh.. but I do have an idea and I know very well what I am saying... You see, this is what it happens when people think God does not answer their prayers, they say, it's God's fault, more problems than solution, I say from the early beginning, God does not answer prayers at all, your faith is the one who manifests the answer through God's power.