Is a mother considered wrong when she uses her knowledge of her doctor daughter to bypass the waiting line at the medical center, considering it "facilitation from our Lord"?

You were right to alert your mother to her mistake, because she took someone else's right to priority. Islam considers priority to belong to whoever arrives first at something permissible. This is based on the Prophet, peace and blessings be upon him, who said: "Whoever precedes to something that no Muslim has preceded to, then he has more right to it." It is not permissible to bypass someone who has priority except with their permission, otherwise it would be an infringement upon their right. Your mother should seek forgiveness from those whose rights she took, or repent to Allah.