Film Theory: The Dark Reason Captain America REALLY Can't Lift Mjolnir
1. Captain America And War
In Civil War, Cap refuses to see the logic of Tony Stark's pleas to fight for the greater good (by allowing Scarlet Witch to be imprisoned temporarily, the Sokovia Accords to be signed and the Avengers to be regulated), because he wishes to be able to do what he wants. In every example of Cap's missions in the MCU, what he wants to do is fight: his increased strength is very much positioned as his power, even more so than his integrity.
Consider what Erskine says about Cap when he reveals why he chose him to become the super soldier. He said that he chose him because he was inherently a good man and that because he had been weak his whole life, knew he would not lose respect for the strength he would gain. In effect, Erskine's lesson to Cap unwittingly led to him valuing his strength over anything else by positioning that as the thing he had to respect.
Cap is a man of war. He is absolutely defined by war, to the point that his important role raising morale with troops in World War II leaves him feeling impotent and also to the point that his "peacetime" activity in The Winter Soldier requires learning how to enjoy himself from a list. It's almost like he's an alien learning what it means to be human.
On a far broader scale, the MCU is about obsession and being unable to give up the addictive draw of heroism. Tony Stark is addicted to being Iron Man to a dangerous degree; Hawkeye and Ant-Man are drawn back to their superhero roles despite knowing it would be better to retire for their families' sakes; Bruce Banner's "hulk problem" is very much a drug allegory and Captain America slots in alongside all of them.
Ultron was right, Captain America NEEDS war. It is what ultimately defines him and he has no choice thanks to what runs through his veins. He is not powered by "superhuman serum" or "superman serum" - it's "super SOLDIER serum". He has been cursed to endlessly live out his duty - to be a weapon even when he's driven underground as a war criminal - until the day he dies.
When he says "I could do this all day," he's not just talking about his commitment to standing up to bullies. He's talking about his appetite for fighting. His capacity for war. And that's why he will never be worthy in the way that Thor is. He has no choice but conflict whether he is unfailingly moral or not.
Everything about Cap's occasionally misplaced idealism - which robs him of the ability to see moral grey areas when they're necessary - is linked to war. His commitment to idealism is very much tied to how he protects those ideals - with the gifts the serum gave him - he's not smarter, he's stronger. His natural response to anything that threatens his sense of moral order is conflict (as in the case of the Accords)