For the Cap theory, maybe he has to sacrifice not Peggy herself, but the idea of Peggy since she died years ago. As he does it Cap moves with his life, no longer being a person trapped in two time periods.
It is possible he is an unreliable narrator, but Red Skull tells Thanos in Infinity War that the Soul Stone "demands a sacrifice ... In order to take the stone you must lose that which you love: a soul for a soul." The implication is that the Soul Stone requires the wielder to sacrifice a literal soul to gain its power.
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Forgot about that, thanks for correcting me.