I'm assuming that generals and admirals are less skilled or prepared to pilot aircraft, as their primary duty is management and leadership, not flying.

Generals and admirals weren't always generals and admirals. As we'll see below, many are very distinguished pilots. And while you're correct that their primary duty is management and leadership, when the general wants to take an aircraft out for a spin who's going to tell them no?

First, let's look at those you mentioned.

General Andrews, USAAF

As has been pointed out in the comments, General Andrews did not die in combat, nor was he believed to be flying the aircraft.

General Worley, USAF

General Worley was a combat pilot during WWII with 120 missions and 215 combat hours, including having been shot down in North Africa. He helped with the transitions to jets. He held the Distinguished Flying Cross, and the Air Medal with seven Oak Leaf Clusters. He died when his recon aircraft was hit by ground fire in Vietnam.

General Fogleman, USAF

General Fogleman is...

...a command pilot and a parachutist, he has amassed more than 6,800 flying hours in fighter, transport, tanker and rotary wing aircraft. He flew 315 combat missions and logged 806 hours of combat flying in fighter aircraft. In early assignments he instructed student pilots, performed combat duty as a fighter pilot and high-speed forward air controller in Vietnam and Thailand, taught history at the Air Force Academy and conducted flight operations in Europe -- including duty as an F-15 aircraft demonstration pilot for international airshows.

He holds the Distinguished Flying Cross with oak leaf cluster, Air Medal with 17 (!) oak leaf clusters, and the Aerial Achievement Medal.

And looking at some modern officers...

Admiral Lescher, USN

The current Vice Chief of Naval Operations of the US Navy is Admiral Lescher. Admiral Lescher was a helicopter pilot. He was a member of and commanded many helicopter strike groups.

Lescher commanded the Vipers of Helicopter Anti-Submarine Light (HSL) Squadron-48, the Airwolves of HSL-40 and the Atlantic Fleet Helicopter Maritime Strike Wing... Lescher’s initial operational tours were with the Lamplighters and Swampfoxes of HSL-36 and 44, deploying primarily to the Middle East region aboard USS O’Bannon (DD 987), USS Clark (FFG 11), USS Capodanno (FF 1068) and USS Elrod (FFG 55)... Ashore, his assignments included leading the SH-60B Seahawk developmental test team as an engineering test pilot at the Naval Air Warfare Center, where he launched the first guided missiles from a Navy helicopter... Lescher graduated with distinction from fixed wing, rotary wing and Naval Test Pilot School training. He has been recognized as the Association of Naval Aviation’s HSL Pilot of the Year, the Naval Helicopter Association’s Regional Pilot of the Year and the Naval Air Warfare Center’s Rotary Wing Test Pilot of the Year.

Admiral Lescher is certainly skilled and prepared to pilot helicopters.

General McConville, US Army

The current US Army Chief of Staff General James McConville...

is a senior Army aviator qualified in the AH-64D Longbow Apache, OH-58 Kiowa Warrior, AH-6, AH-1 Cobra and other aircraft. His awards and decorations include two Distinguished Service Medals, three Legions of Merit, three Bronze Star Medals, two Defense Meritorious Service Medals, three Meritorious Service Medals, two Air Medals, the Joint Service Commendation Medal, two Army Commendation Medals, four Army Achievement Medals, the Combat Action Badge, the Expert Infantryman’s Badge, the Master Army Aviator Badge, the Air Assault Badge, the Parachutist Badge, and the Army Staff Identification Badge.

He is also a quite skilled and prepared pilot of helicopters.

What other GFOs of NATO Rank OF-6 or higher died, while piloting in a war?

I did a quick scan of US officers killed in combat, because that's what I have easily available, and found...

General George Casey, US Army

General Casey hit a mountain while piloting his command helicopter in Vietnam. He was flying partially on instruments. General Casey held the Army Aviator Badge, Distinguished Flying Cross, and 8 Air Medals.

Major General Bryant E. Moore, USAF

General Crumm was killed while piloting a B-52 en route to a bombing mission. He collided with another B-52.

His first assignment was with the 91st Bomb Group in the European Theatre of Operations as a B-17 pilot. He returned to the United States as a member of the "Most Deserving Bomber Crew of the 8th Air Force" and lectured at 30 combat crew training schools and all of the major aircraft factories.

In 1946 he was assigned to the Flying Training Division, Strategic Air Command, as assistant training officer. In rapid succession he became chief of the Bomb Section, deputy of the Training Section and acting chief of the Training Section.

...He attended B-47 Advanced Flying School at McConnell Air Force Base, Wichita, Kan...

The General wears the wings of command pilot. His decorations include the Legion of Merit with oak leaf cluster, Distinguished Flying Cross, Bronze Star Medal, Air Medal with three oak leaf clusters, Air Force Commendation Medal, and the Army Commendation Medal with oak leaf cluster.

Brigadier General Edward Burke Burdett, USAF

General Burdett...

He was commander, 388th Tactical Fighter Wing, Korat Royal Thai Air Base. He departed Korat on Nov. 18, 1967 as pilot of number three aircraft in a flight of four F-105D's, on a strike mission over Phuc Yen Airfield, North Vietnam. His aircraft was hit by fragments. He completed his pass, released his bombs, and made a nearly level right turn to exit the area. His aircraft was on fire. He attempted to light his afterburner, was unsuccessful, and the aircraft went into uncontrollable spin into a cloud undercast. No ejection was seen or parachute observed. He was 18 miles west of Hanoi. He was held to be missing in action from Nov. 18, 1967 to Jan. 15, 1968, at which time sufficient evidence was received to warrant placing him in a captured status. Conclusive evidence was received on April 2, 1974 that he had died in captivity on Nov. 18, 1967.

Awards and decorations: Distinguished Flying Cross, Air Medal, Soldier's Medal, Air Force Commendation Medal, World War II Victory Medal, National Defense Service Medal. Also, he had been awarded pilot wings by the Bolivian Government.