What happened to Jimmy and his family was tragic, and Cas regrets it the most. But on the other hand it's true what Dean told Claire: that thanks to Jimmy's sacrifice Cas could have saved the world. Sure, that was an oversimplification, but it's true that Cas' help was crucial in many cases.
For example, he defended the Winchesters from Zachariah, threw a molotov at Michael, defeated Raphael (Pyrrhic victory, but still), recognized Dick Roman, and exposed Metatron.
He could do all of this only because Jimmy agreed to become his vessel, because that's how angel biology works.
Cas was the only angel who sided with humanity, and he paid a great price for it. And Jimmy gave his life in that fight, but it wasn't Cas who killed him. This wasn't some personal adventure for Cas; it truly was a fight to protect humanity.
Dean told Claire that Jimmy was a hero and that his sacrifice wasn't in vain, and that's the truth. Thanks to this, Claire gained the strenght to forgive. Sure, she still had plenty of problems, but at least this one no longer poisoned her soul.
I'm convinced of that because of Grumpy Cat. Long afterward, she still had it and held it close. If she hadn't forgiven Cas, she would've thrown his gift in the nearest trash can.