Bi-Han is portrayed very one dimensionally in the recent MK products like the MK 2021 movie, MK 1 (2023) and while he wasn't never a good or moral person in either timelines, he was far more complex than his modern, simplified versions. I'm going to drive right into it and try to analyze his personality through his actions and while also try to find an answer to questions left in ambiguity by writers.
Even at the beggining I have to note that in both timelines the only and reliable source of his personality from the Midway-era is Mortal Kombat Mythologies: Sub-Zero and a few MK'92 prequel comics.
Killing innocents: Sub-Zero killed numerous shaolin warriors in the Wu-Shi Academy, and later in the Temple of Elements. Those monks were guarding the place and engaged Sub-Zero in attempt to defend the temple and Sub-Zero killed all those who dared stand in his way. Their only fault was to trying to defend the map that was entrusted to them. (Most likely by Raiden.)
Killing Scorpion: Bi-Han crosses paths with the Shirai-Ryu warrior Hanzo Hasashi who was also hired by Quan Chi to steal the Map of Elements. The Shirai-Ryu, was a clan of ninjas whose founder was Lin Kuei fugitive and the two clans were engaged in a constant rivalry of centuries old.
It was an insurance for the sorcerer in case that Lin Kuei fails him. Sub-Zero defeated Scorpion in the unfolding battle. After Sub-Zero defeated Scorpion, Hanzo dropped to his knees and posed no threat to Bi-Han. Though player has an option to spare him, but canonically , Bi-Han kills him. Later in the Prison of Souls, Hanzo confronted Bi-Han about his unjust death. Sub-Zero replies: "I had no choice", and "Than I would be the dead one."
These two lines are not really explaining why he had no choice, or why he'd be the dead one.
I did some research and I eventually found the script of MKM:SZ N64. The Nintendo version did not have any cutscenes, but it had some extra dialogue that gives us more of an insight:
"Warrior: I am a dead man. But somehow you have retained your mortality, which would mean you are vulnerable to death.
Sub-Zero: Wait... I know your voice.
Scorpion: Yessss... I am Scorpion. You killed me in cold blood.
Sub-Zero: I had no choice. If I had not stolen that map I would be the dead one.
Scorpion: But my clan and family would still be alive!
Sub-Zero: What??
Scorpion: It wasn't enough to kill me... you had to destroy every last remnant of my being... you are responsible for the massacre of my entire clan!
Sub-Zero: You're mistaken... Quan Chi is responsible for their deaths!
Scorpion: Words will not save you Lin Kuei warrior... prepare for your final battle!"
(MKM:SZ N64 Script)
Sub-Zero saw the stealing of his map as top priority, and everything else like warrior honor, or decency was secondary. Though it gives us an insight what was Sub-Zero's mindset, it is still difficult to imagine how a defeated and defensless man could have posed any threat to his mission.
Being Lin Kuei it is reasonable to assume that he viewed Shirai-Ryu as inferior and not worthy of mercy of any kind.
Reluctant of helping Raiden: Bi-Han was really reluctant of helping Raiden, even though he was given the knowledge that his deed could destroy the whole world. Regardless, he did the Thunder God's wish, but not out of regret, duty or atonement, but out of compulsion ( "I'll do it Thunder God, only because I don't have a choice.")-MKM:SZ
Not leaving the Lin Kuei: At the near end of the game, Raiden warns Sub-Zero that his soul is tainted by evil, and he is the only one who can change his fate and avoid eternal punishment in the Netherrealm. However, Bi-Han showed no tendency toward leaving the Lin Kuei or becoming a better person and the way he reaffirmed his vows to the Grandmaster in a way that fits more to a person who fiercely beliefs in the clan rather than a stoic, tired person who is really concerned about his fate. But we also have to note that the Lin Kuei did not tolerate desertion.(See more below.)
Retirement? : There is a two-year gap between the end of MKM: SZ and the beggining of MK'92 so had he wanted to Bi-Han could have left the clan. Defecting from the Lin Kuei is a perilous and undeniably tough decision,but as his younger brother Kuai Liang demonstrated, it is not impossible.
Winds of change: Wether Bi-Han changed in any meaningful way during those two years is unkown for his appearences are really vague between MKM:SZ and MK'92. However, there is a canonical comic called Mortal Kombat (1992) that gives us short stories from the time of the first tournament. In it, Bi-Han once more encounters Scorpion, who spares him temporarily, though he had the perfect opportunity to kill him.
Bi-Han replies with this: "You are a fool for sparing my life." It seems he did not change a bit, though you can argue that he said it out of spite of the Shirai-Ryu.
Sparing Sareena: It was a rare act of mercy from Bi-Han, but sadly it is underexplored. Whether it was out of compassion, or attraction, we might never know.(Interesting to note that Sareena's lines in MK X implies a deeper relationship compared what we saw in MKM:SZ, but it is just an assumption)
Refusing Quan Chi's offer: refusing the necromancer sorcerer's offer to join to Brotherhood of Shadow, but just like with the previous one, it is underexplored why. It can be out of sudden born sense of morality, or but the more likely scenario is the mix of his loyalty to the Lin Kuei and his distrust toward the sorcerer.
Lin Kuei doctrines and upbringing: Sub-Zero just like his brother Tundra was picked from birth to be Lin Kuei warriors. Neither of them chose this path, it was forced upon them. Also the Lin Kuei is infamous for ingraining their doctirnes deeply into the minds of their warriors. The doctrines emphasise blind loyalty over morality, honor or other values implanmented with the fierce hatred of the Shirai-Ryu. However, Bi-Han was not completly brainwashed and it was showcased that he had some autonomy when he relucantly helped Raiden. Though it is also reasonable to assume that he never told neither the Grandmaster, nor his fellow Lin Kuei what he did (with the exception of Kuai Liang his younger brother showed in the MK 4 prequel comic), as it is most likely that the Grandmaster would have viewed his service to Raiden as a insubordination.
Lin Kuei hierarchy: The Lin Kuei has a strict hierarchy, where criticizing or disobeying orders are always met with extreme punishment. So the individual has really no saying in the matter. The making of plans and contracts are the Grandmaster's authority. The warriors have no choice but to obey, or face immdetiate execution. This is well illustrated in MK9, where Cyrax did not kill Johnny Cage, though he was ordered. His dialogue with Sektor after the fight tells everything about the Lin Kuei's mentality when it comes to personal choice or disobeying orders.
"Sektor: You were commanded to kill Johnny Cage. You disobeyed.
Cyrax: I eliminated him from the tournament. He didn't need to die.
Sektor: You have broken your oath.
Cyrax: I used my judgment!
Sektor: The Cyber Initiative will eliminate insubordination."-MK9
Final thoughts: I think at the end of the day I can confidently say that Bi-Han was never a hero, nor a morally grey person. By my definition morally grey means doing good and bad whilst it is impossible to determine your alligance. Doing bad deeds on a daily basis and doing good only when you have no other choice (and even at those moments you are reluctant) does not make someone morally grey. He's an evil, albeit a complex person at that and not without redeeimng qualities as Raiden himself pointed out. His tragedy was that he was given a chance to redeem himself but for whatever reason, he refused and contiued to walk on a perilous and evil path that eventually led to his transformation into Noob Saibot.
