The Bible doesn't say Jesus wouldn't heal those who didn't believe, It says, He Couldn't.
See how many times the Bible says things like, "According to thy faith, and They faith has healed thee".
God has put everything on a faith basis.
The first thing we must know, Is, It is God's will to heal, God not only wants to heal people, But Jesus has already paid the price so we can receive healing, Isaiah 53: 3--5. Matt 8: 16--17. 1 Pet 2: 24.
You mentioned, "Expectation", But while you are expecting, you are putting it of to the future, whereas faith says, "I have it".
Mk 11: 24 and 1 Jn 5: 14--15, says, Faith says "WE HAVE IT", before we see the manifestation of it.
Faith is simply believing that our Heavenly father, [God], Has told us the truth, and receiving the blessings.
So once we have found the will of God, [From the Bible], we can confidently either Minister to the sick, or receive healing ourselves.
God can't operate without man's cooperation of believing.