Yes, I think generally that is true. You need some faith because asking God for healing requires faith to even ask. The more faith the more the result because God is clearly moved by faith as shown in the bible examples you quoted. And, the more persistent the more the result. Think of the parable of the corrupt judge and the persistent woman. The lesser the faith and persistence then the lesser of a result in general. Scripture says that in certain places that even Jesus could not perform any miracles there because of their lack of faith (Mt 13:58).
However, this does not limit God’s sovereign will. If it is God’s will to heal even an atheist then it would be done. Conversely if someone had faith to move mountains but what he was asking was against God’s will it would not happen.
For example, I could have faith to move mountains and ask God for it to rain for 40 days and 40 nights to flood the world in order to get rid of all wickedness on earth, like abortions. We know that such a flood is not going to happen because it would be against God’s will. God promised he would not bring another world wide flood as in the days of Noah. And, such a flood would mean many innocent lives would be killed including good Catholics. So it wouldn’t make much sense. Thus, despite having all kinds of faith it would not happen because it is not God’s will.
If God doesn’t move to heal us, it still may be his will to heal us. But, he is wanting you to mature in your faith or holiness before doing so. Suffering can be a tool for discipline to holiness (see Heb 12:10). Thus, if God was to magically take away all our suffering we could not be purged of our tendency to sin (1 Pt 4:1). And, we could never experience the consequences of our actions like a father disciplining his child would experience. In fact it says that God disciplines those he loves for our own good (Heb 12:6) . And, if we were not disciplined we are not sons but bastards (heb 12:8).
In Catholic teaching suffering is also redemptive. We can offer our suffering up to Christ to unite with his suffering on the cross for others salvation.