When do you give up on a Soldier?

I have a minor difference of opinion of here but I am not an Officer. I give up on a soldier as soon as I conclude further time investment is a lost cause and it's smarter to help them get kicked out of the Army. Why torture the good soldiers in the unit by trying to reform someone that in your heart you know is never going to turn around? Boot them out. That is what the Chapter process is for.

I learned with one of my first roommates in the Army. He was an alcholic. First tried to help him and after being bit or thrown under the bus by him or his behavior more than once. I started to conclude he was not worth salvage. Last straw found out he enlisted with past record of alcholism and a DUI on his record that the Army never found (this was 1984). So that was it and I helped him get out faster by making him look worse as much as I could. Years later I came across him online I think it was 1999, with a Go fund me type website up asking for money. He had been laid off from his job in IT but discussed his Army service and finally at that time he expressed regret for his alcholism. So he did eventually learn his lesson there and maybe getting kicked out helped. I am just happy he was not in the Army all those years with other Soldiers pulling his weight for him.

I was an NCO and cannot relate to the one in your account.

And yes, mission comes before soldiers.
"Mission, Men, Myself" were the priorities we lived as NCOs.

As for the NCO you are leading, I cannot relate to that mindset, trust or no trust.

LT, first you have done a great job thus far, and I can tell you have some leadership experience. My advice is simply this, give this NCO her initial counseling, outlining measured expectations of her as they pertain to her duties and responsibilities as a Noncommissioned Officer. Sir, I further advise you to ask what circumstances led to her not trusting the leadership, what she has done or recommend to improve and rekindle that trust. The advent of COVID (while its a constraint) must not be a continuous scapegoat for the failure to fulfill our responsibilities in any capacity. “Counseling” remains the most effective tool in our leadership Arsenal use it.

Well the mission does come first, but MANY people feel that it's second to the person or a person's career. Perhaps she needs an adjustment.

Never some people are slow to mature and grasp reality . I was that guy . C co 1st recon Cpl Fairbanks crawled head first down the side of a mountain looked me in the eye and called me a slur for a big cat suddenly I didn't fear death anymore I just didn't want Cpl Fairbanks to think bad of me

Another red flag for me is the "trust" comment. Some Soldiers chose not to trust leaders. When Soldiers tell me that they don't trust a Leader, I ask one simple follow up, is your Leader untrustworthy. If a Leader is untrustworthy, I have a path to correct an issue. Stating that a Leader is untrustworthy requires substantiation, I can choose not to trust anyone without cause.

2LT (Join to see) there comes a time when a consensus style of leadership doesn’t fit the situation. You may need to set mission oriented performance standards for the NCO, make sure she understands, and order her to Soldier on and get the job done. Part of being in the Service is learning the needs of the Service are more important than the needs of the individual. That’s part of what separates us from private citizens.

As usual, we are not there and are not familiar with the entire situation, but here are my thoughts. It sounds as though you've attempted to reason; have consulted with the other NCOs and are faced with mission critical situation where your NCO is responsible for training others is not responsive.
Her trust of you is irrelevant. You are responsible for the readiness and training. If you have talked with the other NCOs senior to her and found her response to you consistent with her past performance, perhaps you've come to the point where you need an attention getter.
An Art 15 might be the leadership tact to take. Her performance and attitude indicates she is not ready for her responsibilities. If her past performance merits, a reduction in rank suspended might be appropriate with specific expectations of changes during the suspension period as criteria for retaining her position and grade. If her record indicates less than exemplary performance, a straight reduction would be appropriate.
My point is, you are just starting out, thus involvement of your seniors in resolving an issue is appropriate. All the Art 15s I conducted were preceded by consult with the individual's seniors and a discussion of what steps have been taken to resolve the issue. Your senior may have recommendations of other steps he would recommend, but at least he will know you are recognizing situations and attempting to resolve them. I was once told that all the grades who wore gold insignia are officers in training, take advantage of your seniors who are responsible for helping you through rough spots.

LT. Where is your platoon sergeant?

I know that sounds funny, but I’m actually asking a serious question. As a young LT it seems you might be out in the weeds worrying about managing NCO’s instead of managing the platoon as a whole. This young sergeant needs NCO mentoring not officer mentoring. I am assuming you are her senior rater and not her rater. Give your platoon sergeant a directive to handle it and move on.

If you believe you should give up then they are no longer a military member and should become a civilian again. If they are still worthy of being a service member then find out what is causing their malfunction and help them straighten their act up. Sometimes there will be "turds" that slipped through the cracks and made it into the service. If they cannot or do not have the proper make up for the service then they should be processed out. For the good of the service. As the old saying goes: A chain is only as strong as it's weakest link. Those individuals may end up costing someone their lives.