This would certainly be a good instance. What would the downside be?
In 2 seconds with a file you can change the firing pin signature. You can change the firing pin altogether. For a handgun other than a revolver you can swap out the barrel. For a rifle you can change the bolt and reheadspace. There are to many simple ways to defeat ballistic signature. Plus, if you sell a gun or have it stolen the LEO are still going to look for you if a bullet or case is found that matches your orginal purchase.
Do we fingerprint tires for all new cars purchased? How about taking photographs of every drunk before they leave the bar? How about DNA samples from everyperson born before they leave the hospital? How about manditory chip implants so everyone is tracked by satelite 24 hours a day for their entire life and cameras on every street corner and light post?