I`m not a lawyer so take this advice as only that, not a legal opinion.
First, I`m not sure it is true that a non-US citizen is prohibited from being an officer of a US company or invoved in the day-to-day operations. Two of my partners are not US citizens and do not have H1B visas (they do not reside in the US).
Second, even if #1 above were true, I think you should distinguish between the legal process of setting up the business and the practical matter of running it. As long as the paperwork passes legal muster, you can do whatever you want as far as the day to day operations go. No one will be auditing your day to day activities to determine who is doing what.
If there is some legal reason you cannot be the sole owner of the business due to your citizenship, find a US citizen you can trust to be listed as an officer of the company. Don`t issue any shares, or if you do, issue only a token to your friend or family member in return for the favor, and make the par value rediculously low. Make it clear to your `partner` that once you have your green card or citizenship, you will amend the articles of incorporation and remove them from the list of officers (you may need to buy back their shares if you issue them). Technically you will work for the company as an employee (even if you don`t take a salary) and you can run it the way you want to run it.
I would run this past a lawyer first though, just to be sure you are covered in all legal circumstances.