The Temple sacrifices never competed with the the work of Jesus. The services for each are entirely different, and apply to different jurisdictions, and applications that have absolutely no overlap whatsoever. One (earthly) is patterned after the other (heavenly). One (earthly) is dependent on the other (heavenly). But the existence of one and its services never nullified the existence and services of the other or vice versa.
This is why the sacrifices of animals never dealt with the heavenly requirements of having a circumcised heart, or earning a place in the world to come by one's merit. This is also why the blood of Jesus would have been anathema if offered in the earthly temple, but was accepted in heaven as a payment where the death of the righteousness atones for the world. There is no animal sacrifice that brings one into the World to Come, and the death of Messiah doesn't nullify a Jew's obligation to bring animal sacrifice to the earthly temple for a nazirite vow if a temple exists, one is a Jew, and is in the Land of Israel like Paul the Apostle did and paid for others to do so too at the leading of the Jerusalem Council (Acts 21:24,26).
Bear in mind any theology that deals with determining a "need" for a temple, must be 100% congruent with the following realities:
1. The earthly temple and its services did not exist for thousands of years after the Creation of the World.
2. The earthly temple and its services did not exist for 70 years during the Babylonian exile.
3. The earthly temple and its services continued to exist for 40 years after the death of Jesus.
4. The earthly temple will be rebuilt at some point in the future by King Messiah according to prophecy.
Remember, there are two temples: one in heaven, and one earth made in the pattern of the one in heaven. But their services never nullified each other.
What most people do not realize is that the sacrifices in the earthly temple didn't nullify the work of the heavenly temple, and vice versa. There is no conflict, no overlap. The earthly one allows someone on earth to draw physically near to G-d, especially in this fallen world, where the mere presence of glory of G-d in all his majesty could cause instant death to those not ready. The heavenly one allows one to draw near to G-d on a spiritual level and in the World to Come - and often one's service in the heavenly temple is required before one can even render service in the earthly, else the earthly service would not be accepted. Two different jurisdictions, else there would have been just one temple for all time with no need for an earthly.
I think when people think about the earthly temple, that they completely misunderstand what a gift of G-d it was to a fallen world to be able to literally be before G-d at his doorstep. Granted anyone can pray anywhere, but it is only in Jerusalem, at The Place that G-d has his Name, where we can truly "draw near" to G-d in a real, physical, tangible way, in this life now - and not die. Sadly, it has been taken away from us because of our sins as a nation of Jews - lashon hara, evil speech, sinat onam, unjustified hatred... against the Messiah our King, of all people.