I expect that the insurance companies don't really have much of a say.
It would be impossible for them to examine and risk-assess each and every flight every day and in all conditions, so the only practical way would be for them to cover all flights 'provided they are being operated "in accordance the regulations in force".
It's very unlikely that they have any form of veto.
If heavens forbid something went wrong they might try to recover losses from the authorities _if_ they could prove they were negligent in setting the regulations.
Insurance companies do not have to provide cover. They asses the risk and provide cover with an automatic 10% profit. Engine and aircraft manufactures state that no volcanic dust is acceptable, so as standard no insurance company will cover flying through dust.
So its not so much a veto as an aircraft is uninsurable until its been completely inspected and repaired - probably after EVERY flight. And its not just the engines, volcanic dust screws electronics as well.
Insurance companies cannot blame authorities for opening or closing the airspace, its the airlines who would be ignoring the manufactures guidelines and the airlines a who will bear the cost if they deliberately ignored them.