Paul is wrong. That is basically saying that G-d is tricking us or a liar (G-d forbid) G-d says in Deut 30, regarding the law. (paraphrasing)...This law is not to difficult for you....it is not out of reach... is it up in heaven that it must be brought down? Is it across the sea that we must go and get it? NO! It is near to you!...in your mouth and in your heart that you may do it! Was G-d just saying that? Knowing that it was really a curse? He let his chosen people suffer under the illusion that the law was sufficent, when it really wasnt for 1,500 years from Moses to Jesus? G-d forbid!
The "old testament" people did not get to heaven by doing the law only anyway. Forgiveness (many examples in the OT) is through prayer, repentance, fasting and charity. That coupled with doing ones best to do-keep the mitzvot (commandments)is what gives us a relationship with G-d and forgiveness.
According to Pauls view (or perhaps the church) David would not have gone to heaven. There is not a single case in all of scripture where someone sins and is then going off to do a blood sacrifice for forgiveness! Why? A)Because it is the LEAST effective means of forgiveness (prayer, repentance and obedience is superior) and B) Sacrificial sins were primarily for UNintentional sins. Only about 6 intentional sins could be atoned for by sacrifice (and even then blood is not required, Lev 5:11,12) The rest of intentional sin has ALWAYS been atoned by G-ds grace through prayer, repentance, fasting and charity.
And that is the answer to your question...
Am taking it that your are writing from a Jewish perspective and wanted to know, from that perspective, how sins were atoned for. And you have answered that question.
You replied that it is by "prayer, repentance, fasting and charity". In other words by your own actions.
Now you have not refuted the statement that "G-d will not allow any sin to go into heaven". Therefore, you must be hoping that your "prayer, repentance, fasting and charity" at the moment of death will give you the absolutely clean slate and permit entrance to heaven.
It is the same as the lady who said if you have 'any un-confessed sin in your heart you will not go into heaven'. For in making that statement, she is saying that she too hopes that her 'confession' of any sins on her death bed will allow her entrance to heaven.
Both puts the burden on you that you will know every last sin on yourselves and be able to get rid of them in that very last instance. For if you fail to do so - well, as you know, eternal destiny depends upon it.
However our faith says that G-d is correct in Leviticus 17:11 "For the life of the flesh is in the blood, and I have given it for you on the altar to make atonement for your souls, for it is the blood that makes atonement by the life." we contend that blood is required for every sin down to the last jot and tittle of failure to keep the law - the Absolute law of G-d. We believe that David and all the OT saints were justified through their faith in that symbol of that blood sacrifice.
Which our belief says was accomplish on the cross. And I am sure you have already heard.
But the difference is who it is dependent on. In the case you described and the lady, it is upon yourself and you must do it absolutely perfectly and without error and at the time you are dying.
As for me, I am not perfect and I make lots of mistakes. And I make errors particularly when I am under stress or ill. Not only this, but I have a unconscious tendency to overlook some errors and concentrate instead upon those I consider more urgent. And over a lifetime, despite my digging and trying to root all of them out, and knowing the exactness of the law - I am sure there are enough hiding under the rug - to keep me from heaven if I depend upon myself.
The good news is that G-d provided the first sacrifice "And YHWH made for Adam and for his wife garments of skins and clothed them. which pointed to the ultimate sacrifice of which you know I am referring.
For we believe it is the same faith in what G-d would do, and did, that counts.
Please give careful thought of what I've written as I have of yours. Please think if is possible through your own works at your dying moments to get into heaven? Or is it better to rely on what He has done?