If the Christian prayed at the memorial, then I expect he/she would pray in Jesus' name.
"Would it make more sense to make a prayer that both Jews and Christians can accept?"
No, it would not. Who can or will accept it is irrelevent. A prayer designed for the audience is not a prayer.
JWinNC
This is cure but not true: an important word is implied but missing. A prayer designed solely for the audience is not a prayer.
A minister selecting and formulating a prayer for a particular occasion, although surrounded by his correligionists, designs a prayer.
Any corporate religious experience is collective, hence takes into account those present. A priest at a service for the youth of the congregation, selects and designs his prayers to make them effective for the appropriate age. He does not betray his religion, nor is he lacking deapth.
In every religion there is a personal element and behavior, and there are corporate experiences. Collective prayers are designed for the collective present.
As is said, cute but not true.