Not a historian, but as a law student I feel qualified to give some insight into the reasoning behind the exam.

First, passing the Bar shows you're familiar with your state's laws. As many know, states law vary quite widely depending on where you're at, so having an exam to prove you know the state's law where you want to practice is helpful. If I pass the NY state Bar, that shows nothing about what I know about California law. So seeing that I'm a member of a state's bar lets clients and judges know that I'm competent to give counsel and not overlook a unique aspect of that state's law.

Second, there is an ethics part of passing the Bar. It's not the Bar exam itself, but it's a separate ethics test. It's important to note that legal ethics and moral ethics are totally different. Legal ethics are an artificial construct created to ensure lawyers and judges act in ways that aren't perverse to the goals of a legal system. This is quite different from moral ethics which teach you to act in ways that aren't perverse to the goals of society. Do these two ethics intersect? Yes, at times. But it's important to recognize that they have distinctly different goals.

A lawyer's job is to provide the best counsel as possible, and to represent their clients interests. Lawyers are fiduciaries of their clients. So legal ethics put constraints on how lawyers can act that conflict with those interests. So if an accused murderer tells his lawyer that he did it, legal ethics say the lawyer cannot divulge that because it's not his job to decide if the law says his client is guilty. That's for a jury to determine. So allowing lawyers to circumvent that process goes against the core of our judicial system - it's the people of the community that were harmed by the suspect that get to decide guilt.

So when confronted with that situation, an ethical lawyer might say "Well you don't know if, according to the law, you're guilty of what you say you are guilty of, so let's put that aside." This way the lawyer not only protects the inherent decentralization of power within the judicial system, but also conforms to his fiduciary duties to his clients.

However, if the client said "I did it, and I plan on doing it again" then the lawyer would have an ethical duty to report the intent of his client to commit future crimes (but not the client's confession!). Here, the fiduciary duties take a backseat to the prevention of future harm simply due to social policy. Nothing can be done to prevent the crime that's already occurred, but things can be done to prevent that crime from being committed again by the same individual.

Some might see this as flawed, but it's extremely important to recognize that the judicial system is a careful balance of powers. If one side has weak representation, and the other side has very strong representation, then the adversarial scheme becomes totally one sided and injustice results. This is why we have laws compelling discovery - it's not fair to allow one side to have a surprise witness that the other side can't prepare for and defend against. It's harder to get to the truth when access to information is unequal, so legal ethics are constructed to maintain the balance.

Ultimately, someone could be a total scumbag in their personal life, but still be a fantastic advocate for their client. Legal ethics don't seek to remove these types of people from the pool of attorneys because one's personal moral compass doesn't necessarily mean they're a bad lawyer. It seems counterintuitive, but a hypothetical I always remind myself of is a cigarette company that sponsors a medical study on the health effects of cigarettes. Just because the study was funded by a cig company doesn't mean that the science behind the study is invalid. The same logic applies for "unethical" lawyers. And the opposite is true too. Just because someone is a pastor/priest doesn't mean they're necessarily a good person, even though everyone in the congregation may see them as such.

And finally, it's important to have a standardized system of lawyers. The ABA lets people know who is qualified to practice in the field, while also having a governing body that can set formal, binding regulations on how those in the field practice. Law is a licensed profession, just like doctors or taxi cab drivers. Would you like to see a doctor that hasn't passed his Boards? Or a cab driver who doesn't have a driver's license? Same goes for lawyers. So necessarily, there needs to be a centralized system of governance over the licensing requirements. Sure, in Lincoln's day, nobody passed a Bar and there were competent lawyers. But no one had to get a license to be a doctor back then either, and it's hard to argue that a formalized licensing system didn't help legitimize the medical field.

Hopefully that helps gives you some understanding of the reasons behind why the legal community views the ABA and Bar Exam as important, even though I can't give much insight into the historical creation of the system.

Edit: Fixed some words.

Edit 2: See the post by /u/EvilNalu below for some good clarifications.