Judging from comments, I have been unclear in my earlier posts. Let me try again.
From a purely realist perspective, for the United States, doing nothing trumps arming the rebels.
With a rebel victory off the table for the time being, arming the rebels aims to forestall a government victory and prolong the war. So let’s look at each outcome.
Government victory: What is good for the United States about this outcome? Assad returns much weakened. The chance of violence spreading to Lebanon falls drastically. The fear that rebel factions might be radicalized disappears. The U.S. has no strategic interest in the geography occupied by Syria, so nothing will be denied. (No oil interest, no need for bases, no strategic trade routes, no large-scale investment.)
What is bad for the United States about this outcome? There is some hit to our prestige, seeing as we have (for good reason!) already rhetorically backed the rebels. Hezbollah could claim victory, although given the Lebanese political situation I strongly doubt that they could capitalize on it. The refuge problem would persist after the end of the war.
It seems, then, that an Assad victory would have few costs for the United States. The question then becomes, would a prolonged civil war have an even lower net cost?
Prolonged civil war: What is good for the United States about this outcome? In terms of Assad, the difference is null. He is not a threat while he is fighting rebels, but nor is he a threat if he wins.
Iran is “bled.” I would like to try to estimate an upper bound of the cost of Iran’s involvement, but let’s be wild and call it 1% of GDP. What does costing Iran 1% of GDP get the United States? It doesn’t weaken Iran’s ability to project power in the Gulf. That depends on Iranian ballistic missile strength and naval capability, something not visibly affected by their aid to Assad. Nor does the expense reduce Iran’s influence in Iraq. Finally, it is not enough of a cost to generate substantial domestic unrest. Blogging Bogging Iran down might have meaning were Iranian troops on the ground or were Iran spending a substantial amount of national income, but the first is not true and the second does not appear to be true. The benefit of bleeding Iran is minimal.
Hezbollah is “bled.” The cost to Hezbollah of the war is more substantial, since the organization has limited resources (much of which come via Syria) and has put its own men on the ground. Hezbollah, however, is not a major American security concern ... I would not rank this benefit as particularly important.
What is bad for the United States about this outcome? The big danger comes from the risk that the war will become even more sectarian as it goes on. An increasingly sectarian Syrian civil war risks driving Baghdad closer to Tehran, thereby increasing Iranian power. It also risks having the violence spread to Lebanon, with major risks to the United States.
A prolonged war also risks generating radicalized fighters. Presumably most will remain in Syria fighting the war, but they could generate blowback once it eventually ends.
Finally, the refuge problem will be at least as bad as in the “Assad wins” scenario.
Bottom line: The humanitarian issue is unclear — a victorious Assad might slaughter as many people as a prolonged war will kill — but that is a reason to judge based on cold-blooded national interest. In those terms, there seems to be no discernible net benefit to the United States from prolonging the Syrian civil war. Therefore, there seems to be little reason to arm the rebels.
Avoiding involvement has an orthogonal benefit: it reduces the chance of mission creep and allows the U.S. to stand aside if our preferred side begins to lose. That becomes more costly in reputational terms the more we have invested in the conflict.
Of course, the civil war will probably prolong itself without us. But that is another issue.