When Holy Communion is distributed why is “The Body of Christ” said? Why not say, “The Body and Blood of Christ.”? Or better yet, “This is Jesus.”?

We use the word “body” for several reasons. First of all, Jesus himself used the word “body” (soma in Greek) when giving the Eucharist for the first time: “Take and eat this is my Body (soma).” Hence we, in conformity to Christ, use this way of speaking of the Eucharist.
 
Secondly, both the English word “body” and the Greek word “soma” can refer strictly to the physical dimension of a person, or more broadly to the whole person. In English, I can say, “My arm is part of my body.” Here I am referring to my physical dimension. However, I might also say, “I am somebody,” which refers not only to my physicality but to my whole self. We can also do this in the plural such as when we speak of the “body of believers,” or the “body‐politic.” Here we do not refer to a physical body, but rather to the sum total, or the majority of some group.
 
Hence, we do not exclude any dimension of Christ by referring to his “Body,” as if we were only referring to his flesh. Rather “Body” here refers to the whole Christ. Surely it pertains to a living human body (and Jesus is quite alive) to have not only flesh, but also blood and soul together. We also receive with his body, his divine nature since it is untied to him hypostatically.
 
And while we do speak of what is in the Chalice more specifically as the Blood of Christ, this is only to distinguish its species (i.e. what we perceive) from the host. But once again, Jesus is alive and glorified and his body, blood, soul and divinity are together. Hence, even in the smallest drop of Precious Blood, the whole Christ is received.