I don't think giving up such a valuable and unique weapon just to make such a gesture would be worth it. Civil War was the first movie where Cap does what's best for himself and what he wants to do (helping Bucky and proving he's innocent), and not what's best for other people. I think when Stark tossed the "my dad, who your best friend killed, made that shield, you don't deserve it" card at him, he realized that that's true, and that he isn't worthy of what the shield symbolizes anymore.
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I don't think giving up such a valuable and unique weapon just to make such a gesture would be worth it. Civil War was the first movie where Cap does what's best for himself and what he wants to do (helping Bucky and proving he's innocent), and not what's best for other people. I think when Stark tossed the "my dad, who your best friend killed, made that shield, you don't deserve it" card at him, he realized that that's true, and that he isn't worthy of what the shield symbolizes anymore.