Very funny. Even in 2015 we still rely in many ways on the old good SMS. No, operators are not making the money on the service as they were used to in the past. However it is not easy for them to fully replace the service (e.g. by RCS). That’s why the VoLTE standard GSMA IR.92 says:
The UE must implement the roles of an SM-over-IP sender and an SM-over-IP receiver, the IMS core network must take the role of an IP-SM-GW.
Update: SMS is about to stay even in 5G. So much for RCS replacing SMS.
In other words, the VoLTE network has to support the (legacy) SMS sent over IP (SIP). The VoLTE phone will receive a common (binary) SMS and the native client will display this message as any other. The only difference is that this time the SMS is sent from an IMS network over SIP protocol. Mind the purpose is not just to support common text messages, but also to support OTA messaging for (U)SIM provisioning, SMS ‘non-text’ applications or Message Waiting Indication for Voice Mail.
The network functionality which provides messaging service in the IMS network is called IP-SM Gateway (IP-SM-GW) and from the IMS point of view it is an Application Server. IP-SM-GW Call flows originate or terminate in LTE network and IP-SM-GW forwards the messages towards/from legacy SMSC, which is still located in CS network.