I agree with most of your first paragraph, the only thing that I think it's important to point out though is that Libya really wasn't Obama's war, the US took a backseat on that one to the British and the French. But yes, I agree that current Syria policy is largely informed by the disaster that is Libya, and that in general the administration's MENA policy is a resounding failure.
As far as strategy goes, I think that the administration's actions in the region are largely determined by goals, not strategy. I don't think that the US actually has any unified strategy, I think that there are four conflicting goals.
Keep oil flowing from the gulf. This is probably the most important of the goals, the global economy could not survive the consequences of a sustained crash in gulf energy exports.
Nuclear non-proliferation in the region. I don't think that the US foreign policy establishment cares all that much about whether or not Iran dominates the region, I think that most people would not like to see that but are willing to trade the region to Iran in exchange for a nuclear deal.
Preventing terrorism from establishing a foothold in the region. This is why the US pretty much abandoned its anti-Assad goals, there was a realization that groups like ISIS and JAN are more destabilizing to the region and dangerous to the US than Shia dominance or tyrants. And yes, the lessons learned from Libya play heavily here.
Protect Israel from annihilation. A lot of people see a conspiracy here, or AIPAC or whatever, but ya know, there are many states whose protection the US sees as core interests. We give less aid to Israel than we do to Japan or South Korea, we have mutual defense assurances with the NATO countries, and we regularly intervene violently at the request of Yemen and Pakistan, we went to war in 1991 for Kuwait, etc... The American relationship with Israel is very controversial and it raises a lot of emotion, but I don't happen to think it's particularly special. We will let Israel take rockets from Hezbollah or Hamas but we would not allow the state to be overrun, which is pretty much the same relationship that we have with Japan and South Korea. Right now this means very little in practical terms because Israel's neighbors are all fairly weak or friendly.
So yeah, I think that if you view US policy as being invested in these goals, it makes much more sense. The Obama administration doesn't have a strategy but it inherited a few core goals that have been longstanding fixtures of the American interest in the region.