Firstly I think you need to prepare yourself for the fact that you probably just didn't get the job - while recruitment processes can drag out (especially at large companies) hearing nothing for months after an initial application is usually an implied rejection. Yes it's a bit rubbish that they don't send even boilerplate rejections but it's still hugely common.
That said there's no real harm to be done by following up as to whether the position (or a similar one) is still open. Unless the position you applied for is at a very senior level however I don't think messaging the Executive Director is the right approach though. Chances are he won't even know about the position let alone that you applied or what happened to your application.
Your proposed message would be like sending the President of the US an e-mail asking why your local city councilor hasn't gotten back to your e-mail about street lighting from 6 months ago. They just aren't the right person to ask as that information simply isn't in their orbit.
A better approach would be to reach out to HR or to whomever you directed your original application to and politely inquire with them. Doing this is unlikely to cause any harm to any future chances there.