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

Without a doubt. You're older now, probably more mature, and may even find that the extra 4 years may afford the oportunity for that promotion. Of course your Commander will more than likely want to get a committment from you, but you can't beat the benefits. Don't pass up an oportunity of a lifetime over a missed promotion. I was passed over several times and I didn't let that get in my way. I just dug in harder to make sure it didn't happen the next time. You have to remember there is one slot that is going to get filled with that promotion and there are probably three or four others that are competing against you for the same oportunity. Make sure you shine above all the others, keep your head above water and if you stumble get back up and do it all over again until you get iot right. Your day will come.

If he was Discharged at 16 yrs, there are many other factors that come into play before making the decision.
Short answer, YES.

Join the Reserves, volunteer for deployments to get activated under Title 10, add that time to your active duty time and get over the hump of 17-years of active military service, then ride out the last three years to get to 20. Of course, all of this only works to get you a pension assuming you chose the traditional retirement selection when the DoD changed the retirement options.

At a minimum you could go into the reserves for 4 years, retire and start collecting retirement pay at age 60.

I almost got out after 10, but I stuck it out and stayed for my twenty. I am so glad I did. It led to other good things when I retired and now I have three retirements. Get your butt back in and do that twenty. You might even find things are to your liking now and stay for more. Good Luck

Why not?

The simple answer is yes. The best answer depends on the individuals circumstances.

My question is did he have a chance to go down in rank to an nco and finish his 4 years?

I had a RIF Major in my AIT class in 1989. He reenlisted and was a SGT. Good man. Didn't know what happened after he left school.

Yes. You retire at the highest grade earned. You can do four years standing on your head. It’s awkward running into folks who used to work for you or who you graduated from school with. But they don’t cash your checks and pride shouldn’t prevent you from getting the retirement you earned.