Why can't God forgive a sinner without having to kill Jesus the righteous one?

Other respondents speak of God's justice and humanity's debt, which is part of the answer. A vital reason has to do with the definition of love. God acts justly, but "God is Love". (See 1 John 4:8) Thus God is the definition of love. So what does love do?

Greater love has no one than this, that someone lay down his life for his friends. (John 15:13 ESV)

I am the good shepherd. The good shepherd lays down his life for the sheep. (John 10:11 ESV)

The ultimate act of love is to give your life for your friends. God goes beyond this by giving his life for his enemies to make them friends, if they will have it.

6 For while we were still weak, at the right time Christ died for the ungodly. 7 For one will scarcely die for a righteous person—though perhaps for a good person one would dare even to die— 8 but God shows his love for us in that while we were still sinners, Christ died for us. (Romans 5:6-8)

If a person thinks that it is beneath God to die for his creatures, they are right, but to do things that are beneath you is a sign of humility. False religions have proud gods. Christianity has a humble God.

28 Come to me, all who labor and are heavy laden, and I will give you rest. 29 Take my yoke upon you, and learn from me, for I am gentle and lowly in heart, and you will find rest for your souls. 30 For my yoke is easy, and my burden is light.” (Matthew 11:28-30 ESV)

One final observation. Firefighters die rescuing strangers from buildings. Police die fighting criminals to protect their communities from evil. Doctors succumb to illnesses that they contract from their patients. Soldiers die defending their homeland. If giving your life for your friends is the highest form of love and God did not do that, it would mean that some humans are more loving than God. It would mean that some of us are more righteous than God. That cannot be! So since we have these examples of virtue among us, God, who is greater than man, must also give His life for His friends.

The early church father Anselm might have provided the best answer for that in his "Faith and Reason."

As he reasoned it out, man could not be one with God because of his sins (God cannot be one with sin), but man was incapable of paying for those sins. Therefore, God, whom the sins were owed, became man, and as a man paid the penalty for us so that we could be one with him.

Now, its still a brain teaser on how this works. The perplexity of the problem is beyond our understanding, but we know it works.

At it's heart sin is a disposition of rebellion which produces behavior that incurs debt. The debt of actions may be forgiven as OP suggests and yet the disposition remains:

Therefore is the kingdom of heaven likened unto a certain king, which would take account of his servants. And when he had begun to reckon, one was brought unto him, which owed him ten thousand talents. But forasmuch as he had not to pay, his lord commanded him to be sold, and his wife, and children, and all that he had, and payment to be made. The servant therefore fell down, and worshipped him, saying, Lord, have patience with me, and I will pay thee all. Then the lord of that servant was moved with compassion, and loosed him, and forgave him the debt. But the same servant went out, and found one of his fellowservants, which owed him an hundred pence: and he laid hands on him, and took him by the throat, saying, Pay me that thou owest. And his fellowservant fell down at his feet, and besought him, saying, Have patience with me, and I will pay thee all. And he would not: but went and cast him into prison, till he should pay the debt. So when his fellowservants saw what was done, they were very sorry, and came and told unto their lord all that was done. Then his lord, after that he had called him, said unto him, O thou wicked servant, I forgave thee all that debt, because thou desiredst me: Shouldest not thou also have had compassion on thy fellowservant, even as I had pity on thee? And his lord was wroth, and delivered him to the tormentors, till he should pay all that was due unto him. So likewise shall my heavenly Father do also unto you, if ye from your hearts forgive not every one his brother their trespasses. - Matthew 18:23-35

The Son of God became flesh in order to neuter the disposition within us and give us a new disposition, not of flesh (wherein dwells sin) but of spirit (wherein dwells life):

Let not sin therefore reign in your mortal body, that ye should obey it in the lusts thereof. Neither yield ye your members as instruments of unrighteousness unto sin: but yield yourselves unto God, as those that are alive from the dead, and your members as instruments of righteousness unto God. For sin shall not have dominion over you: for ye are not under the law, but under grace. What then? shall we sin, because we are not under the law, but under grace? God forbid. Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness? But God be thanked, that ye were the servants of sin, but ye have obeyed from the heart that form of doctrine which was delivered you. Being then made free from sin, ye became the servants of righteousness. - Romans 6:12-18

I would like to add a bit to Anne's answer above.

Rom 3:21-26 But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets; Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference: For all have sinned, and come short of the glory of God; Being justified freely by his grace through the redemption that is in Christ Jesus: Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.

The death of Christ brought out a righteousness that is beyond that of the law. To put it in perspective:

Righteousness is right actions. The law is the standard of Righteousness - it tells what is right and what is wrong.

The law puts in a demand and gives a guarantee. The demand is:

Matt 22:37-39 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. This is the first and great commandment. And the second is like unto it, Thou shalt love thy neighbour as thyself.

The guarantee is:

Lev 18:5 Ye shall therefore keep my statutes, and my judgments: which if a man do, he shall live in them: I am the LORD.

So demand is to love the Lord with all heart and your neighbour as yourself and the guarantee is that you will live if you do it.

Jesus loved us more than He loved himself - because He laid down His life for us. Thus exceeded the demand of the Law. Exceeded the righteousness of the Law. However He died for us - the law guaranteed life but this did not happen with Jesus.

Because of this act of Christ, God has gained another righteousness which is apart from the Law, the righteousness which is by the faith of Christ. This righteousness allows God to be just and yet justify the sinner. This is one of the reasons why Christ had to die.

Hope it gives a different perspective of the same issue.