So could Stanley McChrystal have just sent the 60,000 troops he wanted without a presidential directive to increase the troop level? Why or why not?
Legally? Yes - so long as McChrystal had the additional forces at his disposal in his unified combatant command, which I'm sure he did, and many more. McChrystal was the unified combatant commander. He as full operational control of the troops attached to his command, both in the theater and in garrison. Where did that authorization come from? It came from the Authorization for Use of Military Force Against Terrorists (AUMF) passed in 2001. This authorization predates Obama. With or without Obama, McChrystal has operational control until such time that he's either replaced, or he's ordered to cease hostilities.
But, as a practically matter, no. Just like for all practical purposes, it would have been impossible for Bush to have sent the additional troops to Iraq absent at least minimal support from Congress, although it's clearly within the President's prerogative as CiC to send troops virtually anywhere - at least for short periods of time. Why is that? Because Congress controls the power of the purse strings. So, while the President is the CiC, the Congress feeds, clothes, arms and pays those troops. Without support of Congress, there is no war and there are no additional troops, in any theater.
Obama, Gates or McChrystal could have ordered the deployment, but Congress wouldn't have to pay for it, ergo there would be no deployment - such is the net effect of the there separate, but equal branches of government.