Ok, doing some careful thinking on the fly. Don’t try this at home.
Obviously it depends. They should act exactly the same with reference to the law. They should act the same with reference to the generic fruits of the Spirit, e.g., showing mercy, seeking peace, feeding the hungry, clothing the naked and so forth. I think you are asking, “Does the family/Church order laid down for men and women in the Bible extend to the civil realm?” Even that appears to me to be a “it depends,” with the question resting on heart motivations. Let me paint two pictures.
Woman A has always chafed under whatever male leadership she’s experienced. She longs to lead to finally show men that “women can do X just as good as men.” She works her way to the top of an organization, working long hours, doing objectively excellent work, and obtaining promotions in the process until she is recognized in her field and in charge of men. This is sin, but traced back to its root one can easily discover the sin of Pride. Even if this woman were removed from a position of formal leadership, she would remain miserable and in her sin, for she was acting out of a heart that refuses to bow the knee - both to God as Father, or to any man, in any context, for whatever reason.
Woman B submits happily to male leadership in her Church and home. Let’s assume she is young and unmarried, and shows a particular aptitude in a STEM field, for instance. She has a teachable heart and a brilliant mind. As she works through college, she develops something with public marketability. It is her idea/product and she gives God the glory for it. She decides to move to market with this product and finds herself running her own startup company with both male and female employees. Eventually she gets married and has a child, deciding to stay at home with the child, but unavoidably maintaining some amount of decision-making power in her company which is in caretaker status.
While I realize I may be brushing up against a straw man, I’m not sure how else to frame it. These two archetypes do happen in various ways. I would find the idea that Woman B violated God’s law by exercising her God-given ability to the best of that ability in the civil/commercial sphere; that she sinned by not finding a male to run the business for her - I find this idea unsupportable in and foreign to Scripture.
So to answer your question: It’s not a question of shouldness but isness. Men and women simply will be different in all sorts of contexts, attack problems differently, strategize differently, assume risk differently, resource things differently, relate differently. Whether these differences inhibit or facilitate the task at hand in the commercial/civil domain, is the it-depends.
Where you sit is where you stand. I’m not a pastor, I’m not a welder, I’m not a construction worker. All I can go off of is my experience in the military and DoD. I’ve had great female bosses and terrible female bosses. I’ve had great male bosses and terrible male bosses. The good bosses shared certain qualities across the board that I would not associate with masculinity or femininity. They include truthfulness, clarity, organization, timeliness, urgency and fairness. The bad qualities also are not associated with masculinity or femininity: they include insecurity, incompetence, dishonesty, complacency and hypocrisy. I have never run into sexual tension over reporting to a woman as another commenter suggested would be inevitable. Perhaps that is because of the military emphasis on mission execution. Perhaps it would be different if I were fitting pipes, I don’t know.