I’m very accepting of other’s beliefs (or try to be), but I’ve found over the years that I’ve had to be clear about what distinguishes their beliefs from mine. I also had to notice that others’ statements about Jewish belief (or statements between Jewish groups about each other’s beliefs) were sometimes (if not often) factually mistaken. So, I offer the following as a clarification of one area that divides Christian from Jewish belief, without in any way denying the sincerity or validity of Christian devotion in its own terms.
A friend who is “Messianic Jewish” told me that he didn’t understand “how good works and prayer can be a substitute system for a temple system that required blood sacrifices?”
His question was based on the Christian scripture: “Without the shedding of blood, there is no remission of sins.” 
My answer to him included the following:
The idea that “blood” was “required” is, I’d have to say respectfully, much more of a Christian idea (specifically Paul’s) than a Jewish one,  notwithstanding:
“For the life of the flesh is in the blood; and I have given it to you upon the altar to make atonement for your souls; for it is the blood that makes atonement by reason of the life.” 
That G-d requires of us something more than “blood” — especially the blood of perfunctory sacrifices — goes back to the prophets, even to the psalms, in verses like “The sacrifices of G-d are a broken [i.e. humble] spirit;” “…do justly, love mercy, walk humbly with your G-d;” etc. On the basis of these verses, and others like them, the rabbis taught what they did.
But they never actually taught that these things “replace” sacrifices. During the time before the destruction of the 2nd Temple, “good works and prayer” were required along with sacrifices. So — there was no thought at that time of replacing sacrifices. The devotion that was the essence of sacrifices (or should be) was to be carried into our daily activity, in part by “good works and prayer.”
When the Temple was destroyed, the question was asked, “How can we atone for sins?” Also — “How can we perform the sacrifices that are required of us, if the only place that we could do the sacrifices is no longer functioning?” The rabbis answered by saying, in so many words, that in the absence of the physical sacrifices, we could still continue to show our devotion by doing “good works” (living a life of devotion and obedience to G-d through doing the “mitzvot”). The formal requirement of sacrifice could even be fulfilled by our reciting the relevant passages from Torah.
Any sacrifice that involved “blood” could also be done with grain/meal/flour (i.e. with no blood at all), if the person bringing the offering was too poor to bring or purchase an animal. In such cases, the offering was no less “acceptable.” So, “blood” was never understood to be the essential element of the sacrificial system. Blood was smeared on doorposts, but only once, on the night of the Exodus.
It’s an interesting topic, and one that I could probably write more about. Put simply, the “Christian” understanding of Judaism isn’t necessarily the “Jewish” understanding of it. That’s what’s giving rise to confusion, I think.