Split roles
Let's think about what would happen if the president and commander-in-chief (CiC) roles were separate. What roles go with what?
Defense. Pretty obviously the current Department of Defense would go to the CiC. Otherwise, what power does the CiC have?
Taxes. This would seem obviously the president's. But if the president controls how much money the CiC gets, then the CiC is subject to the president? If the CiC has separate taxes, do they get a separate Internal Revenue Service (IRS) too?
Research and development. If this gets split into military and civilian parts, both could. But then both could blame the other for insufficient spending on their pet projects. For example, NASA (National Aeronautics & Space Agency). The military has used it to launch satellites and work on missile rocket technology.
State. If the president keeps this, then the two can blame each other for foreign policy failures. Why hasn't the military solved this pesky problem? Meanwhile, the CiC could say that this is a diplomatic problem. And of course other countries would have the opportunity to play upon those divisions. If the CiC takes this, then it exacerbates the previous problem.
Foreign Aid. Obviously this goes with State, except where do they get the money? If the president keeps State, does the CiC have the ability to veto military and technological aid?
NSA. The National Security Agency has a foreign focus that is meant to inform both diplomatic and military actions.
Intelligence. Some intelligence gathering is to prepare for the possibility of military action. Some is to advance diplomatic concerns. Some is to detect domestic problems like terrorism.
Homeland Security. This is for domestic protection. However, Homeland Security exists as a department because the FBI and CIA were deliberately separated. But terrorists don't have rules against coordinating between their foreign and domestic parts. Homeland Security has to operate partially in foreign areas. And it coordinates with Defense to target foreign terror cells that support domestic terrorism.
Coast Guard. The Coast Guard operates close to domestic shores. But it is far more like a military agency.
DEA. The Drug Enforcement Agency attempts to interdict drugs from entering the country. And it enforces laws domestically. And it coordinates with other agencies, like Homeland Security and the Coast Guard.
Other law enforcement. Alcohol, Tobacco & Firearms, the Secret Service, and the Federal Bureau of Investigation also have to coordinate with each other and previous agencies.
Presidential protection. Currently provided by the Secret Service and the military. Air Force One is a military plane. Marines protect the White House. OK, let's give all that to the president. Who protects the CiC? Is there a separate apparatus for that?
Energy. Obviously domestic right? Except that the Department of Energy is responsible for providing nuclear materials for nuclear bombs. An obviously military responsibility.
Commerce. Again, sounds domestic. But a significant portion is commerce with other nations, foreign affairs.
Veterans Affairs. It matters more to the military, but it's a domestic responsibility.
There aren't natural splits for these roles that work in all situations. And this isn't all of them, but I'm tired of typing. If the president is the CiC, then the president can balance these roles. If they're two separate politicians, then they will joust politically between them for power.
Splitting the roles is a recipe for dysfunction.
Blurring the lines
Also look at what Barack Obama did to subvert the separation of powers. As a candidate, he decried use of executive orders to implement policy. As president, he created an entire bureaucratic regime in Deferred Action for Childhood Arrivals (DACA) by executive action. Consider what a commander-in-chief could do:
Want to implement a basic income? Start a mandatory militia program with compensation but no responsibilities.
Want to implement single payer health care? Expand the Veterans' Affairs department to civilians. Or create mandatory military enrollment so that everyone is a veteran.
Want to implement free college? Again, turn everyone into a veteran so that the GI Bill applies.
Presidents have similar opportunities:
A president could expand armed groups on the domestic side. Perhaps the protection detail. Or the coast guard.
As seen in free trade agreements, domestic regulation affects international relations. However much is given to the CiC, a president will want that back. Increasingly, domestic and international issues are intertwined.
Coups
If you give one person all the military power and someone else all the domestic power, you increase the chance of a coup. As the military gets more and more frustrated with the split, they have the ability to take power for the CiC. Also, the separation of roles means that the CiCs are likely to be associated with the military. After all, who else is qualified?
Unified tickets
A lot of the focus here has been on what can go wrong if the president and CiC do not coordinate. But what if they do? We could have President Trump and CiC Michael Flynn now. And there goes separation of powers.
Separation of powers and the consequent checks-and-balances of the system work because there is a natural tension between the branches of the government. Most Supreme Court justices were appointed by previous presidents, so they don't feel beholden to the current president. Legislators represent smaller districts than the president's national district, so some of them will be against the president's party. But the CiC has the same district as the president. The same voters choose both. Nothing separates them.
To maximize their power, parties would make deliberate attempts to win both positions at once. And they would sometimes succeed. Even if the elections are staggered, note what happened in 2006 and 2008. The Democrats won in both, taking control of the presidency, Senate (with a 60-vote super majority), and the House of Representatives. There seems little reason to believe that they would have been satisfied with just one of president and CiC.
History
Historically, the main job of the president was related to foreign policy. For example, George Washington, Andrew Jackson, and Ulysses S Grant were all admirably qualified to become commander-in-chief. For that matter, presidents like the Adamses, Thomas Jefferson, and James Monroe had diplomatic experience before becoming president. James Madison and Martin Van Buren were two of the five Secretaries of State in the first eight presidents. Tenth president John Tyler was the first president with only domestic experience while eleventh president James K. Polk was the first elected president with only domestic experience.
The twentieth century shifted the presidency to a more domestic role, only Dwight Eisenhower came from a top military position. George H. W. Bush was the only president with ambassadorial experience since James Buchanan. But the United States system was designed with the idea that diplomacy and military were the natural responsibilities of the federal government while domestic authority was at smaller levels.
Source: Wikipedia
Even if we admit that modernly the roles could be separated (and I don't), it would not have made sense when the roles were being designed. Presidents were primarily commanders-in-chief then.
