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Captain America is put in every situation Law was put in, in the movie The One. He has to either do what Law did or outperform him. If he fails, the results are obvious (death). No shield here. What happens? If he needs a gun, he can get one in the situations Law needed them.

Cap dies in the opening scene lol... he'd get shot up while in that vent

Then I doubt he'd be able to outrun the cops but its possible. Though he might guess Roedecker's trap and escape it though.
It is unlikely Yulaw's girlfriend would rescue Cap but even if she does, Cap won't be able to escape since he doesn't have the gunplay skill or the future tech knowledge Yulaw has
Second Swat scene repeat outcome of the first, though the cops lose the advantage of Gabe sensing their opponent.
He might be able to escape them
Without the resemblance Gabe will not be framed so he''ll get help so Cap will have to run. Also due to lack of resemblance he won't be able to fool TK
He will lose to Gabe too
He can stomp Rodecker and Funsch though, maybe even easier since he won't be arrogant like Yulaw

It's no escaping. Cap has to take out everyone Law took out. If Law escaped, Cap gets to escape as well. If a fight took place, Cap has to fight it out as well.

He fails