Unity through Service Act of 2021 This bill establishes a Council on Military, National, and Public Service in the Executive Office of the President to, among other things, advise the President on promoting and expanding opportunities for military service, national service, and public service. The council shall establish an internet-based platform to serve as a centralized resource and database for people to learn about and connect with service organizations and opportunities, and assist in the recruiting needs of service organizations. The council may carry out a pilot program to focus on recruiting individuals from underserved markets and demographic populations, such as those defined by gender, geography, socioeconomic status, and critical skills. The Department of Defense, the corporation, and the Peace Corps shall establish a joint plan to provide an applicant who is ineligible, or otherwise not selected, for service in the Armed Forces, in a national service program administered by the corporation or in the Peace Corps, with information about forms of service for which the applicant has not applied.