"What assurance do the current and future operators of this aircraft have that the problem is fully understood and has been thoroughly designed out to prevent a recurrence?"
Given that France, Germany, Spain and UK all released their aircraft for flight following the initial investigation, these operators must have received suitable assurances from Airbus Mil or the aircraft would have remained grounded. As an example, Boeing assured the UK that Rivet Joint was fully airworthy at point of delivery but the MAA was not convinced; the aircraft stayed on the ground (for many months) until the UK had enough information, beyond that provided by the manufacturer, to satisfy the assurance requirements of the post Haddon-Cave airworthiness environment. The fact that any assurance in this case has not been made public does not mean that none has been offered to customers.
As for whether it was a civil or military flight (question from atakacs), most nations treat the operation of aircraft designed for military purposes as being military in nature, regardless of the status of the crew, because the aircraft are not designed or built to the standards that would apply to a civilian aircraft. It would be for Spain to decide on publication of the investigation report for a military accident that happened on its turf, not Airbus.