To me the term "flying aircraft carrier" implys something analogous to a ship, but in the air, not just a big airplane which carriies a parasite fighter. The vehicle must have an endurance significantly greater than normally provided by a conventional airplane or ground effects craft - and be capable or servicing and (to at least a limited extent) repairing the aircraft it carries, launches, and retrieves in flight. I suppose the endurance problem for an HTA craft could be solved by providing continuous air-to-air refueling or a nuclear (or solar?)powerplant, but it is hard to imagine an airplane large enough to provide service and repairs facilities for a squadron of aircraft. Perhaps a surface effect craft could do this, but I really don't consider these true aircraft.
As others have said, only the USS Akron and USS Macon, as well as the aborted ZRCV designs, qualify as true flying aircraft carriers, and the utility of airships in this role would become limited as performance and stalling speeds of the airplanes it would carry increase. The carrier planes would have to match the speed of the airship to be hooked and brought aboard. Since the theoretical maximum speed of a large rigid airship is barely 100mph, this would limit you to piston-engined planes. No high performance jets. Thus the weapon system would meet its technological dead-end by the end of WW2, if not before.
As to how "flying aircraft carriers" might become the only type of aircraft carriers, you would have to significantly retard development and acceptance of airplanes by at least 15 years or speed up the development of airships by that same amount. Perhaps, in this TL, if the efficiency and utility of large rigid airships as the key naval scouts had already been proven, and 10-15 years of operational experience had clearly defined their roles, it is possible that airplanes, when developed, would be seen naturally as adjuncts to airships - not competitors. Airplanes would be added to increase the airships' scouting abilities and provide some sort of limited stand-off offensive capability. Perhaps, just perhaps, this might eliminate consideration on investing gobs of dollars in a "new technology" such as airplane carrying ships.
This is highly unlikely, however, since so many of the technologies which made effective rigid airships possible the early 1900's (development of cheap light alloys and light internal combustion engines) were the same developments which helped make the modern airplane possible. Once it is appreciated that airplanes could carry bombs and torpedos and be launched from real ships, there is no way an aircraft carrying airship could match the offensive punch of even a small full-deck carrier. Plus it would have nowhere the endurance of a surface ship. What the airship gains by its greater speed (80 kt vs 30 kt), it loses by its greater vulnerability to weather conditions. Even if it can't operate its planes, a ship can stay on station during all but the worst storms and be ready to resume operations as soon as weather permits; an airship would either be confined to base or forced to leave its assigned station when weather was risky or dangerous.
The only long-term role for a airplane carrying airships would probably be in ASW warfare