Is it not possible to tension such a structure that the whole thing is ridged, like a steel truss, and can carry at least it's own weight, regardless of support, or would this be considered wasteful for the end requirement?
How rigid? If you mean zero deflection - that's nearly impossible to achieve. Every bridge structure has some deflection.
Generally, structures are designed using a computer aided design program and loads are simulated and stresses analyzed through finite element analysis. Programs like AutoCAD have many add on modules like Nastran, the FEA module, and several others. Depending upon the work done by the engineering firm they may also use specialized analysis programs like Dlubal that provides analysis and design of beam structures, concrete structural members, steel and aluminum structures, timber structures, glass structures, connection analysis, dynamic stress analysis, and tensile membrane structures.
The point being - you can design the entire structure and the stress analyze every member and the entire structure. This includes static loads, dynamic loads, wind loading, lifting points, support points, etc.
I'm sure the NTSB will be going over the engineering documents and CAD analysis to determine whether the design, construction, or movement procedure were at fault. But, it is hard to believe that the engineering firm did not design the bridge with a safety factor as engineers are, by nature, conservative and philosophically use a "belt and suspenders" approach. Engineers are well aware of their responsibility in providing safe designs and their liability if something goes wrong.