Short answer is no, airships would still have died out without the Hindenburg tragedy. It was only the last in a long line of disasters reaching back to the First World War, where the fragile nature of rigid airships was exposed.
The biggest enemy of airships wasn't fire, but rather weather. An airship is quite fragile for its size, mostly because there is only a limited amount of aerostatics lift available for whatever volume of lifting gas you use, ad every extra ounce of structural weight in the airship reduces the amount of mass you can carry as payload. Suddenly encountering turbulence or storm fronts would be very bad for the airship, and airships generally had (or would have even today) low performance, so would have a difficult time getting away from bad flying weather.
This also translated into difficulty in ground handling. Airships on the ground are still lighter than air, but being attached to mooring masts or ground cables will become sails in any gusting wind. Controlling an airship on the ground is quite difficult and time and resource intensive, far more so than wheeling an airliner up to a terminal and having the passengers just walk on or off the airplane. Airships were easily damaged in ground handling accidents, which now means a very expensive asset needs to be repaired and is not available to earn revenue.
Finally, although airships were huge vehicles and required equally huge investments in infrastructure (look at illustrations of the airship hangers built during the 1930's, for example), their payload was actually quite limited. Airliners even of that time could be amortized much more quickly because they could carry more payloads per time period, and of course modern airliners can carry as much or more than any historical Zeppelin or rigid airship, and travel at high multiples of an airship's speed. Crossing the Atlantic in hours vs crossing in days is a winning business proposition for air transport.
Airships can still fulfill niche roles which would be difficult for other types of aircraft, such as the surveillance aerostats which can hover and "stare" at particular zones with sensors for periods measured in days or weeks. One can probably still make a case for larger versions of WWII era blimps for anti submarine patrols, or to fly over task forces and carrier groups to provide airborne sensor coverage for long periods of time, but obviously helicopters and fixed wing aircraft can do the job well enough, because there are no serious proposals that I am aware of for recreating blimps in these roles.