Probably the best, recent book to tackle this topic is Bruce Longenecker's The Cross Before Constantine: The Early Life of a Christian Symbol (2015).

Longenecker argues, successfully I would say, that the cross as a Christian symbol emerges prior to Constantine. This is an argument against a prevailing position that it was entirely post-Constantine. His argument does not discount that its usage was both transformed in significance, and greatly popularised, in the Constantinian and post-Constantinian age. However, based on material and some textual evidence, he points out the use of the cross in the 3rd and into the 2nd century.

In particular, he notes the apotropaic use of the cross - as a symbol to ward of evil/supernatural powers. This is an important dimension not to be neglected.

As to your main question, I would argue that the reason is primarily because of the theological import that the crucifixion has among early Christian texts. In this, it might be considered parallel to the use of the title "Good Friday" for the day of crucifixion - surely it was anything but good? And yet, in the context of (early) Christian it is precisely the death and resurrection of Christ that effects atonement for sins, and the subsequent defeat of death itself. It is precisely this reversal of the shame and dishonour of the cross into the central theological event of Jesus' life/death/resurrection, that causes the cross to become a symbol of Christianity.