The sacrificial system was a mere shadow of the true sacrifice, Jesus Christ. This is why the animals used must have no blemish or spot, because they represented Christ. As John the Baptist said,
The next day John saw Jesus coming toward him, and said, “Behold! The
Lamb of God who takes away the sin of the world!"
Jesus takes away the sin of the world. The sins of those that were before Jesus were forgiven in credit, and they were to look forward towards the Messiah who would come and truly pay for their sins. Many forgot this and simply made it a routine.
We look back at the one who paid our sins and look to sin no more as we know what it cost.
Those who have trusted Jesus to take their sins, and have walked as He walked, will be raised in the last day, to everlasting life. This is why many of the texts you provided speak about the last day and why Jesus and Martha have this conversation:
Jesus saith unto her, Thy brother shall rise again.
Martha saith unto him, I know that he shall rise again in the
resurrection at the last day.
Jesus said unto her, I am the resurrection, and the life: he that
believeth in me, though he were dead, yet shall he live:
And whosoever liveth and believeth in me shall never die.
Believest thou this?
She saith unto him, Yea, Lord: I believe that thou art the Christ, the
Son of God, which should come into the world.
Mary, Martha, and Lazarus were very close friends of Jesus. He visited them often. He taught them many things, and it was obvious that this is one of the things He taught them. This is also what He taught others.
“When the Son of Man comes in His glory, and all the holy angels with
Him, then He will sit on the throne of His glory. All the nations will
be gathered before Him, and He will separate them one from another, as
a shepherd divides his sheep from the goats. And He will set the sheep
on His right hand, but the goats on the left. Then the King will say
to those on His right hand, ‘Come, you blessed of My Father, inherit
the kingdom prepared for you from the foundation of the world...'"
Whether they lived before Jesus, or after Jesus, those who believe on Him, repent of their sins, and then walk according to the will of the Father, when they die, they sleep until the coming of Jesus where they will be resurrected unto everlasting life.
EDIT: After reading the comments, I realize i should be more precise. According to the Bible, only a very few number of people are alive in heaven right now. These are Elijah (2 Kings 2) and Enoch (Genesis 5:24) who did not die, but were taken by God; Moses (Jude 1:9), who died but was resurrected by God and taken to heaven; and probably the people who resurrected with Jesus (Matthew 27:52-23) as witnesses.
Everyone else, like David, is "sleeping" awaiting the Second Coming.
"Men and brethren, let me freely speak unto you of the patriarch
David, that he is both dead and buried, and his sepulchre is with us
unto this day... For David is NOT ascended into the heavens: but he
saith himself, The Lord said unto my Lord, Sit thou on my right