The early church understood this, and not just in the era of the Apostles. Throughout the first two centuries, wherever Christian witness went, physical healing went too. It would perhaps be helpful to mention a saint or two who healed, but the fact is, healings may well be the most common miracle ascribed to those the Roman Catholic Church honors as saints, famed and forgotten saints alike. While it became much less common after the time of Constantine, it did not vanish like most other miraculous acts and signs. Instead, nearly every era has some Christian who acted as a faith healer, right through to our own times.
Healing is a sign of the kingdom, of the day when all illness and unease is healed.
It is actually one of the actions of that kingdom asserting itself, the beginning of the healing process for all of creation. It is no substitute for the completed Kingdom. The healed will eventually die, just as Lazarus died again. In healing prayer and healing work, we are a part of God's holistic healing work. Christians don't worship health.