No one is falling over themselves to respond to Barack Obama's quest for a new "coalition of the willing" to attack the jihadis of the Islamic State in Iraq and Syria – least of all, it seems, the Arab governments that are most immediately threatened by its brutal, border-demolishing agenda.

The dangers are not in doubt: Jordan has been suffering the jitters since Islamic State (Isis) fighters took the northern Iraqi city of Mosul in June. This week it announced the arrest of 40 alleged extremists as a "precautionary measure".

The Saudi authorities have rounded up "sleeper cells" said to be recruiting terrorists. The normally taciturn Saudis have been vocal in denying they support or finance Isis, and insist they abhor its extremist and revolutionary ideology – even though their own penchant for beheadings is often cited by critics as one unpleasant likeness.

Elsewhere in the Gulf, the UAE has made clear it considers Islamists of all hues a threat, at home and abroad. Even Kuwait, probably the biggest single source of private funding for extremists fighting in Syria, has cracked down. It and other states that looked the other way when their citizens funded jihad against Bashar al-Assad are now cooperating with the US Treasury department. Qatar has cleaned up its act as well.

The Gulf's limitless financial resources alone were never enough to guarantee a coherent effort in the war against Assad. Bandar Bin Sultan, the last Saudi intelligence chief, was adept at delivering cash to rebel groups, but there was too much competition and not enough strategy and control.

The problem with the Saudi-Qatari effort, argues Bruce Riedel, an old CIA Middle East hand and director of the Brookings Intelligence Project, was that it did not have an equivalent of the ISI, the Pakistani intelligence agency that ran the Mujahideen war against the Soviet Union in Afghanistan. The Gulf sheikhs were no match for Iran, Russia and Hezbollah, determined to defend their ally in Damascus.

Conventional military capacity, especially in the air, is not a problem. The six members of the Gulf Cooperation Council have hundreds of advanced combat aircraft.

Turkey, Jordan and Egypt have hundreds more. Saudi pilots have flown sorties against Houthi rebels in Yemen, the Emiratis in Afghanistan, and now against Islamist targets in Libya, a sign perhaps of a greater readiness to act without US leadership.

But problems of command and coordination persist. "It is hard to see how the UAE air force or any GCC air force could provide any real firepower complement to US airstrikes beyond the symbolic legitimacy of an Arab state participating," said Fredric Wehrey, of the Carnegie Foundation.

The case for action against Isis is that the Sunni states have a responsibility to help defeat a group that has a warped sense of its own legitimacy and has behaved with such horrifying cruelty that it has been dubbed "al-Qaida on steroids".

In the trenchant words of the Iraqi Kurdish commentator Hiwa Osman: "Isis is a Sunni Arab problem. Neither Kurds nor Shia can end them. Sunni Arabs inside and outside Iraq should cooperate to do so."

But there are reasons for reticence, too. The growth of sectarianism, linked to the strategic confrontation between the Saudis and Iran, has fed an already strong sense of Sunni solidarity: thus Gulf fury at the Shia-dominated Maliki government in Iraq, which did so much to promote an anti-Sunni agenda. The hope is that the new Baghdad government, headed by Haider al-Abadi, will prove more consensual.

"Each of these states, whether they like it or not, is bound to Iraq and Syria's warring factions by tribal links, religion and history," said Wehrey. "Their rulers are still sensitive to public opinion and especially the pockets of pro-Isis sympathy among certain segments, some of them wealthy and influential."

Another worry is that fighting Isis will inevitably mean strengthening Assad, as western calls for tacit cooperation with him suggest is already happening. Bombing the Arab Sunni heartlands of Iraq and Syria is not the same as action against the universally despised Muammar Gaddafi in Libya in 2011.

Last weekend the escalating crisis prompted an emergency meeting of Arab foreign ministers in Jeddah, the summer retreat of the Saudi government on the Red Sea. No decisions were announced but the emphasis appeared to be on increasing coordination and exchanging intelligence to stem the flow of cash and fighters flocking to the black banner of the Isis "caliph", Abu Bakr al-Baghdadi, and on backing the new Iraqi government. Doing more than that may be more than the fractured Arab system can bear.