Your question is flawed in its premise.
“Who runs the battlefield” is too generalized. The President is the Commander in Chief and Congress has war powers as well.
The President, through the Executive Branch decides the overall strategy of the war...however, Congress may restrict Executive powers (See War Powers Act).
The Congress is authorized to generally run the military and regulate the military, however. This comes into play in this situation. The President has dictated ROE to the Generals, who follow the Presidents’ orders.
However, the Congress has every right as stipulated in the Constitution to write law to right something, they as Congress, see as fatally flawed. The current ROE is fatally flawed.
Divided government has peculiar strife which was forseen by the Founders. The Supreme Court has for the most part ruled that they will not decide these issues and will let them play out in the political arena. In other words, the Executive and Legislative will have to fight it out.
...or you Paulistas can jump out of the warm bath of your fantasyland and look at reality.
http://thomas.loc.gov/cgi-bin/bdquery/D?d112:18:./temp/~bdulyM::
No, it's straight to the point. You want 536 CINCs, whereas the Constitution stipulates 1, and for good reason.
The President, through the Executive Branch decides the overall strategy of the war
Do you think Congress is responsible for the tactics? When, in 220+ years of our government, has the Congress dictated ROE to the CINC? Congress declares wars, Presidents wage wars.
This comes into play in this situation. The President has dictated ROE to the Generals, who follow the Presidents’ orders.
You should keep in mind that the current ROE were suggested by the generals, as part of the COIN strategy, so you're actually suggesting Congress should override the generals and the CINC in this instance with respect to strategy and tactics.
However, the Congress has every right as stipulated in the Constitution to write law to right something, they as Congress, see as fatally flawed. The current ROE is fatally flawed.
The Constitution stipulates how the CINC can be replaced if you don't approve of his strategy and tactics (we hold these events every four years), but it doesn't subordinate the CINC's authority over the battlefield to Congress.
And since you ducked it again, I'll post it again: If Congress set a restrictive ROE, would that supersede the orders of the CINC, in your opinion?
It isn't generalized at all. It is the essential function of the CINC.