Executive search firms need a good resume from you, but they also must have a search assignment that matches your background before they can be of service. Headhunters do not market you. They find the best candidates for client job openings. Their clients are companies that pay their fee or retainer. Don't expect a headhunter to make you her number one marketing project. Conducting your search campaign isn't her job, it's yours.

No one can help you in holding a good job except Old Man You.
Edgar Watson Howe

There is no universally preferred format. Many job seekers spend days perfecting their resumes, agonizing over whether to use a chronological format or a functional one. This question may be akin to figuring how many angels will fit on the head of a pin. There is no one perfect resume to suit every employer's needs. But there is a perfect resume for a specific opportunity. If you are going to focus on perfection, do it on an individual, rather than a global basis.