Not quite, although you are close.
SMS messages get sent through the GSM control channel that normally handles the control plane traffic (e.g. call sent-up/tear-down, base station hand-over, et al). This implementation came about due to a historical oddity; basically when the GSM standards were first defined they discovered that they had some spare capacity left on the control channel so someone had the bright idea of defining a messaging service although "no-one would ever want to use it" (famous last words IMO); this is way SMS messages have such an odd maximum size - it reflects the message size of the GSM control channel.
You are right in that SMS transmission is more-or-less "free" to the network operators, their only costs is in maintaining the SMSC and the inter-operator links.
Biting the hand that feeds IT
