No, "car" is an exceedingly reasonable word.
It's not, because it meant something else - reading 19th century literature is somewhat mentally confusing...
You simply aren't going to get people to use phrases or long words in every day speech. "Horseless carriage" is too long. That's as likely to get shortened to "car" (possibly pronounced "care") as it is to anything else.
Most likely to "horseless" (cf. wireless), or maybe HC.