Might I posit to you that the death of Yeshua was not meant to replace the sacrificial system, and the early believers did not see it that way either? I believe, and there is good evidence to support this belief, that the early believers did not view the sacrifice of Yeshua as a replacement for the temple service as has been taught by many Christians. If they believed so, why did they continue going to the Temple, move to Jerusalem to be near the Temple, continue making sacrifices as outlined in Acts and other scripture, and continue to do so up until the destruction of the temple in 70AD - around 30 years after Yeshua ascended. In fact, the evidence points that up until the destruction of the Temple that we could even say the early believers were a Temple sect. I believe the scriptures are quite clear that once the Temple is rebuilt that the sacrifices will continue, and be performed by the Aaronic priesthood again, while overseen by the Messiach. This is actually a messianic prophesy that both Christians and Jews can agree on. Perhaps they believed that the sacrifice of Messiah is to atone for sins, but that is not a replacement for all of the different sacrifices that were called for by Hashem, most of which were not for guilt.
Non believers continued to go to the temple. The disciples continued to go to the temple, for that is where the Jews were gathered and their mission was to the Jew first. The believers themselves were gathering in houses and riverbanks. There is not one hint in the NT that animal sacrifices of any kind were to continue. For it is Jesus Himself that is now our High priest, it is He that offers all grace.
Mat_12:7 But if ye had known what this meaneth, I will have mercy, and not sacrifice, ye would not have condemned the guiltless.
Eph_5:2 And walk in love, as Christ also hath loved us, and hath given himself for us an offering and a sacrifice to God for a sweetsmelling savour.
Heb 4:15 For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin.
Heb 4:16 Let us therefore come boldly unto the throne of grace, that we may obtain mercy, and find grace to help in time of need.
Heb_7:27 Who needeth not daily, as those high priests, to offer up sacrifice, first for his own sins, and then for the people's: for this he did once, when he offered up himself.
Heb_9:26 For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself.
Heb_10:5 Wherefore when he cometh into the world, he saith, Sacrifice and offering thou wouldest not, but a body hast thou prepared me:
Heb_10:8 Above when he said, Sacrifice and offering and burnt offerings and offering for sin thou wouldest not, neither hadst pleasure therein; which are offered by the law;
Heb_10:12 But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God;
Heb_10:26 For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins,
Heb_11:4 By faith Abel offered unto God a more excellent sacrifice than Cain, by which he obtained witness that he was righteous, God testifying of his gifts: and by it he being dead yet speaketh.
Heb_13:15 By him therefore let us offer the sacrifice of praise to God continually, that is, the fruit of our lips giving thanks to his name.