I can understand the meaning of the phrase off to hell..., but I was wondering why, of all the possible vehicles that may have been chosen, it came to be in a handcart?
This is mainly due to the alliteration of the phrase:
"Going to hell in a handbasket", "going to hell in a handcart","going to hell on a Harley", "going to hell in a handbag" and '"sending something to hell in a handbasket" are variations on an American alliterative locution of unclear origin, which describes a situation headed for disaster without effort or in great haste.
You can see that all of the objects above begin with the letter h. To say "going to hell in a VW" or "going to hell in an ice cream truck" would have less impact.
Pithy sayings of this sort often involve either alliteration or rhyme ("In like Flynn," "wake and bake"), which give them a tag-like quality that's easy to remember and rhetorically more emphatic.
As a child, I remember rag and bone men in England in the 1950's would push a handcart. There were different designs of these contraptions, but mainly in the north of England they were pushed rather than pulled. However whether pushed or pulled, the progress of the rag and bone man was very slow as he stopped at each house. Hence, going to Hell slowly (in a handcart). For interest, I remember the call of the rag and bone men, it was almost musical.