To allow the sender to set the TTL, right? Without adding another field to the packet header.

If you count up from zero, then you'd also have to include in every packet how high it can go, so that a router has enough info to decide if the packet is still live. Otherwise every connection in the network would have to share the same fixed TTL, or obey the TTL set in whatever random routers it goes through. If you count down, you're always checking against zero.