Yes, general board or senior-executive oversight is compatible with executive capacity. The role still must direct management, establish goals and policies, exercise broad discretion and primarily perform executive work.
Explain the actual governance relationship
Show what the board approves and what the beneficiary decides. A signature authority or executive title alone does not demonstrate the full role. If the board directs everyday operational tasks, explain that accurately so counsel can assess whether the duties fit.
The foreign employment and qualifying-business requirements remain separate parts of the petition, even where the proposed U.S. governance structure is clear. Reporting to a board is ordinary corporate governance and is contemplated by the standard, which refers to receiving only general supervision or direction from higher-level executives, the board of directors or the stockholders.
What matters is the character of that oversight. Establish how often the board meets, what it actually considers, whether it reviews performance against a plan or directs operational decisions, and what the executive may do between meetings without asking. Then look at the minutes rather than the description, since minutes record what a board does and are difficult to reconstruct.
A board that approves a budget and holds the executive accountable for delivering it is overseeing; one that decides which customers to pursue and which staff to hire is directing, and the executive in that arrangement is implementing rather than exercising wide latitude.