No, that's not correct. Legal Aid will not represent where a person is earning too much, has too many assets, or has a case they deem to be an unlikely success (usually where orders have already been made by the court, etc.).
Family law matters are not decided based on which is the more capable parent. They're decided based on what's best for the child. Your argument should always be focused on what the child has to gain from spending time with you, rather than what the child might lose from spending time with the mother. Basically, you have a better chance of success arguing that "I'm a great parent", rather than "she's a bad parent".