The sources on the crucifixion are fragmentary and contradictory. But even if an archeological discovery proved the Jews responsible, it shouldn’t make the slightest difference in Christian-Jewish relations.
For centuries, some Christians projected their anti-Semitic biases onto the events surrounding the crucifixion. They imagined the Jews set out to kill Jesus because he was a Christian, and only a people that hates Christians could do that. They imagined the Jews set out to kill the Son of God, and only demonic people could do this. Because of this, countless Jews were persecuted as “Christ-killers”, and Crusade upon Massacre upon Pogrom were perpetrated against the Jews.
This was a mistaken approach, because it reinforced the myth that somehow Jesus was a Christian cast into a foreign Jewish society. In actuality, the crucifixion occurred in the aftermath of an intramural Jewish struggle, involving theology and politics; all involved were proud Jews. These sorts of internecine battles sadly still occur today, in events like the assassination of Yitzchak Rabin.
The Jew vs. Jew context of the crucifixion narratives render the question of who killed Jesus irrelevant to Christian-Jewish relations. Vatican II recognized this. Pope Benedict XVI continues to recognize this. It’s a tragic shame that Christians in previous generations couldn’t recognize this; far too many Jews lost their life due to a theological fraud, an anti-Semitic misreading of ancient Jewish history.