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

You need to look at the Officer separation regulation and the regulation that covers eligibility to enlist/reenlist. I researched this for a similar question a while back and I remember reading that prior service officers were ineligible. But can not rember whether it was RC or AD applicable.

If they can enlist (reason for discharge may determine this) for the last 4 years then go for it. They'll receive their pension at the highest rank attained.

Every time, don't give up.

ahhh,
of course you should.

If they are ELIGIBLE...which MOST are NOT. During the drawdown in the 1990s, ONLY OCS officers WITH TIME LEFT on their enlistment prior to OCS could reenlist. ROTC & USMA grads were out the door. Mebbe things / policies have changed?

Yes

I believe he should be allowed to re-up to finish his time to retirement.

What an absurd question. After 16 years what's 4 more?

It's hard to know the circumstances or record of this officer, but any member who serves 10+ years is crazy if they don’t continue with another service, in the reserves or some other method. I have seen this mistake that everyone regrets when they get older. There are too many benefits that all of us have earned and most do not use or obtain when they get older. Complete the career (20) and move on with life.

Absolutely. It would be ridiculous not to.