Forget humanitarian arguments. US policy is almost entirely driven by an obsession with Man-portable air-defense systems (MANPADs) falling into the hands of 'terrorists' and 'jihadis'. It would rather every single Syrian die than facilitate the spread of these weapons - after all, if a US or Israeli aircraft was blown up by one, the president might lose the next election!
At this point, there's little to worry about. The FSA has SAM-7s, a Soviet-era model which, in long experience, has proven just about useless against fixed-wing aircraft. By now, even commercial airliners either have or can acquire countermeasures against these devices. But without a no-fly zone, the FSA will doubtless do its utmost to acquire more capable MANPADs. It's unlikely they could get the US Stinger, but presumably there are more advanced Soviet/Russian systems available. Serbia and Iraq have had one such system, the SA-16, for years. The Tamil Tigers acquired one and it saw service in El Salvador and Angola. It has a good record even against fairly advanced aircraft. Commercial airliners would be sitting ducks.
The implications are clear, at least on the assumption that Turkey and Jordan could be persuaded to cooperate. If the US does not establish a no-fly zone, that greatly increases the probability of what it fears most, the proliferation of effective MANPADs. If the US does establish a no-fly zone, it greatly diminishes that probability. Perhaps this is obvious to American intelligence organizations. But since US policy shows no sign of intelligent direction, it seems worth pointing out.