Does the 3 years of service requirement for officers include time spent as an enlisted service member?

Yes, prior enlisted time counts towards the 3-year service eligibility requirement.

Officers with a minimum of 8 years prior enlisted Active Duty service (Army, Navy, marine, Air Force, Coast Guard) including Limited Duty Officers (LDO) and Chief Warrant Officers (CWO) are eligible to use TA or NCPACE at all officer ranks. All other Officers become eligible upon promotion to O-3

NOTE: This article is in reference to NAVADMIN 114/19 and NAVADMIN 112/23