I think Rosecrans was sincere. Like McClellan, he saw the war through the lens of his own position, and found great difficulty in placing the national interests above the interests of his own command.
Strategic considerations had Washington doing everything they could to make sure Grant took Vicksburg. They wanted Rosecrans to pressure Bragg to keep him from detaching against Grant. Rosecrans refused.
Rosecrans waited until Bragg had dispatched 10,000 reinforcements to Joe Johnston, trying to build a relief force for Vicksburg. Tactically, this was very good for Rosecrans and his effort. As things worked out, those 10,000 were in transit during the Tullahoma Campaign, and did not arrive in Mississippi in time to take part in an effort to relieve Vicksburg.
In hindsight, it doesn't matter. But in real life, hindsight is what doesn't matter.
If Pemberton held on longer (a definite possibility to consider in June 1863) and Stevenson's 10,000 reinforced Johnston (they were en route) and Joe Johnston does attack successfully, disaster threatens Grant. That's what Lincoln, Stanton, Halleck, and Grant wanted to prevent. That's what Rosecrans refused to do anything about. Rosecrans was in the wrong, no matter what his local situation was, because he was unwilling to subordinate himself to higher command.
Regards,
Tim