There are communications and communications. Sometimes an important person has to speak with an important person or with a group of listeners; this the computer can handle, allowing the Captain to speak his piece and the enemy Captain or the mayors on the cities of the planet below to listen. No need for a top-level officer for operating the comms in that case.
Sometimes, though, constant messages have to flow back and forth multiple starships to keep things happening in proper order. For this, one needs a number of people, each sending out his or her own message as needed, and listening as needed. But that's by definition several people, not one top-level officer operating the comms.
I don't know what the role of the comms officer in TOS really was. Connecting the calls didn't seem technically demanding; perhaps Uhura or Palmer not only connected the calls, but also carried out constant communications with various parties - the HQ, nearby ships, the various departments of the Enterprise, scheduled reports, timely announcements, what-have-you - and we never noticed because they had those fancy earphones and throat microphones and whatnot and we didn't hear a sound. Would the need for that go away if tech advanced even further? It's very difficult to see why the tech wouldn't have been as advanced in the 2260s as it eventually was in the 2360s - even today's computers ought to manage that sort of communications complexity with ease, after all. But perhaps Comms is a major department with several "speakers" belowdecks to keep in touch with the several parties involved, and some skippers want that department to have a spokesperson up on the bridge while most do not.
In ENT, the communications officer is also a linguist and a contact specialist of sorts. Perhaps most starships have a linguist, but some have this officer belowdecks, some on the bridge, and some combine the role with the comms department head and either have that on the bridge or belowdecks...
Timo Saloniemi