Did factors specifically related to the concorde design influence the Air france flight 4590 crash?
In short: No, the crash of 4590 could happen to other aircraft in the same situation: i.e. a blown out tire, a tank fire, subsequent structural issues, and at least two engines (of 4) not running properly.

The official accident report summarizes the causes pretty well (and is worth a read)


  The accident was due to the following causes:
  
  • High-speed passage of
  a tyre over a part lost by an aircraft that had taken off five minutes
  earlier and the destruction of the tyre. 
  
  • The ripping out of a large
  piece of tank in a complex process of transmission of the energy
  produced by the impact of a piece of tyre at another point on the
  tank, this transmission associating deformation of the tank skin and
  the movement of the fuel, with perhaps the contributory effect of
  other more minor shocks and /or a hydrodynamic pressure surge. 
  
  • Ignition of the leaking fuel by an electric arc in the landing gear
  bay or through contact with the hot parts of the engine with forward
  propagation of the flame causing a very large fire under the
  aircraft’s wing and severe loss of thrust on engine 2 then engine 1.


None of these are related to the design or construction of the airframe itself. This podcast with one of the pilots also touches upon the accident and is perhaps one of the best primary sources of Concorde info.