Is the statement: "O Allah, this is not a covenant with Your saints" at the hospital door an expression of discontent, despite acknowledging one's sins? And does the fact that the son was not admitted to the hospital, by Allah's grace, negate that?

You should not have uttered that phrase because it implies wonder, presumptuousness about one's deeds, and impatience for relief. A servant, no matter what calamities befall him, should know that they are due to his sins and shortcomings. He should humble himself before his Lord, admitting his wrongdoing and shortcomings, saying: "There is no god but You; exalted are You. Indeed, I have been of the wrongdoers." It is better for a servant to approach his Lord through the gate of humility and submission to Him, glorified be He.