First, thank you. I used to go back and forth with my Bible study leader over this. His answer was very much the same as yours, and I didn't buy it then, and don't buy it now; here's why.
What happened to the sacrificed animals after the priests kicked everyone out of the outer holy? That explanation also doesn't answer the question of why God required sacrifice at all. If God knew that animal sacrifice didn't quite have the "juice" to absolve those sins as a system of law that was nearly impossible to follow 100% introduced, then we've got ourselves a nifty system of control, here.
If the Old Testament saints were looking forward to the Messiah, then why were the Pharisees hellbent on killing him? They demanded he be put on the cross, not God.
I think most agree that the way one got into heaven back then is the same as now. The Old Testament prophets may have done some things like sacrificing to the priests the best of their stock (also important to realize), but how they entered paradise (heaven wasn't for saints, yet) was simple. Live a good life in faithful servitude to your fellow man. Just like Christ told us to do.
Now, again, why does God require a sacrifice? Let me rephrase: Did God ever really require a sacrifice at all? I feel the real truthful answer to this question is going to make some people's heads explode, because it means we're really dealing with something way more convoluted in terms of Biblical history.
Old Testament saints got into Paradise by living their lives "as Christlike" (before there was a Christ/Messiah example); sacrifice or not. So, who was really demanding all those sacrifices? If I didn't need to sacrifice my stock to the priests to gain entry into heaven, then why was it "demanded by God"? Why did "we" stop sacrificing to God after Christ? Again, he said he didn't come to abolish the law, but to fulfill it. Christ did not instruct us to stop following the law. Christ instructed us to approach following the law from the perspective of a heart filled with love instead of legalism and judgement. Big huge difference, there.
Theory:
All God has ever required of us is our love and fellowship with our fellow man. Take all Jesus' teachings, and there isn't anything in there regarding sacrificing animals and wheat. The sacrifice Jesus spoke of was of yourself. Point me to scripture that shows what/when Jesus sacrificed while at the temple?
How much of our Biblical history is shrouded in deceptive teachings? What religion do we know of thrives off sacrificing life in exchange for "something"?