The meaning of serving HaShem out of Bitachon, says the Holy Gaon of Vilna, (in his commentary on Mishlei) is to trust in HaShem without any intellectual crutch.
What exactly is an intellectual crutch? To my limited knowledge it would seem at first that any belief other than Ain od milvado (there is no other) would constitute a rational crutch. To even say HaShem created the world and the system of health and recovery, and naturally I should therefore trust in Him as the Creator of sickness and health, that He can and will heal someone.
At first this would seem like a crutch, the need to go into the reasoning, even a little bit, is a step away from pure bitachon, pure faith and trust. In a way it may seem tainted.
Yet there is a deeper level. Once you understand that HaShem is all that there is, and HaShem created and continues to create existence on every level. Then it can be said that through the understanding that HaShem created also medicine, sickness and health that we are drawing down our awareness of HaShem into these understandings, these relationships as well.
Through drawing HaShem down in this way, we are crowning HaShem in our intellect, in our ego, recognizing that our entire understanding of reality is for the sake of HaShem and our closeness with Him. We are endeavoring to connect with HaShem through our every thought, through our every understanding.
This is the deeper level, but there is first a more basic level. Our yetzer hara, our evil urge, may try to dissuade us in our service of HaShem through bitachon and emunah. he tries to distract us and convince us that what is on the outside is more real and more true than what we know to be on the inside. When we explore what is before our eyes, a sick person,(has v'shalom) a doctor, medicine, and we begin to remind ourselves, "HaShem is the reason behind the sickness," "Hashem is the reason behind the doctor's abilities, knowledge, and success," "HaShem puts life and healing in the medicine, just as he puts life in the food and drink before us, in the air around us." Each reminder, turns the object before us from a vessel for the yetzer hara into a vessel for Holiness.
May it be the will of HaShem that His nation will be healed speedily, may it be the will of HaShem that His words, "I am HaShem, your healer," will ring with certainty and truth in our ears. May it be the will of HaShem in His infinite mercies that we merit to know Him in all of our thoughts, and recognize Him in every element of our world.
כל העולם לא נברא אלא בשבילי - the whole world is created in my way [according to my actions]. Yet we know that HaShem created the whole world for His Honor. It is our job to ensure that the world that our actions create brings Honor to Him and not to us.
May HaShem grant Rav Mordechai Eliyahu long days and years to reveal HaShem's Honor to all of Bnei Yisrael.
אין עוד מלבדו