Veterans Application Assistance Inefficiency Decrease Act of 2021 or the VA AID Act of 2021 This bill requires the Department of Veterans Affairs (VA) to implement a pilot program to assess the feasibility and advisability of providing claim enhancement assistance to individuals who are applying for certain VA pensions in order to reduce the backlog and expedite processing of claims. Specifically, the program applies to those applying for service pensions, non-service-connected disability pensions, or pensions to surviving spouses and children. The Government Accountability Office must complete a review of the pilot program and a study on the feasibility and advisability of permanently extending claim enhancement assistance to individuals and spouses applying for the specified VA pensions.