Should an Officer who was non-selected for promotion, and discharged at 16 years of service, reenlist to finish their 20?

Absolutely, 4 years is nothing compaired to the 16 you've already served. Why would you give up a retirement check? (Reserve retirement may not pay the bills under the old system, but it is time served and money earned.)

The short answer would be yes. But what are your current circumstances? Are you disabled, reenlistment code, age etc. If you are receiving a VA disability and was accepted by a branch for reenlistment you would have to suspend your disability payments or your military pay, you cannot receive both. Are you thinking Active or Guard/Reserve, officer or enlisted? Way to many variables to give you a straight answer. My suggestion if you’re serious about pursuing this are contacting a recruiter and seeing what options you may have. Good luck!

Short answer: yes. As I understand your profile, you are a Captain in the Reserve. If the question is not hypothetical, but is personal, you've been twice non-selected to major. Your profile shows you were previously an E-6. If you choose to enlist to complete your 20 good years, then you will probably come back in as an E-6.

My experience with officers going to enlisted to complete 20 is limited. I worked with a captain pilot who was twice passed over, but was allowed to remain on active duty as an O3 to 20 years because the Air Force was short on pilots. See if your MOS is critical. I also was a Scout Leader with an Air Force SSgt (E5) who had been a Captain. He sued the Air Force, the officers who wrote and endorsed his last few OERs before his promotion board. He won! He went to work one morning as a SSgt and came home that afternoon as a major. This is rare. I don't recall the exact grounds for his suit. It was not because he was discriminated against as a member of a protected class. It had something to do with supervision and geographical separation from his rater. There was an operations clerk at an Air Base where I worked who was a Master Sergeant, but wore Command Pilot wings. I asked about him and was told by my boss he had been a major who was twice passed over for lieutenant colonel. He chose to finish out his 20 as an enlister person then decided to stay even longer.

I think it's worth it financially. You'll probably retire as a Captain with 20 years of service. Good retirement pay. You'll be eligible for TRICARE till you're 65, then TRICARE for Life. Your retired ID card will get you installation access and use of AAFES, Commissary, Pharmacy, etc. It's well worth it.