Depends on the quality and design of the PSU, and the failure mode. If something internal has shorted, dragging the power rails down, then you may be OK. If something failed and the output over-volted AND it has over-voltage protection, you should be OK. Otherwise
Certainly an option for determining if there has been any other damage. As for the shuttle PSU... personally, I'd open it up (usual caveats - make sure all high energy components are dischaged, only if you know what you are doing and are comfortable working on equipment that can kill you if you don't know what you are doing and see if there is anything visible - popped capacitor would be a possible culprit.
Then (if you are sufficiently skilled and you can find something obvious) you could replace that component. Then test the PSU carefully to make sure nothing else has gone, and job done. But there are a number of ifs and buts there.