There are two rivers in this conversation and they are not interchangeable. The Toccoa above Lake Blue Ridge is a shallow, rocky, mostly Class I mountain river that you float on an inner tube with a drink in your hand. The Ocoee, forty-five minutes or so north-west across the Tennessee line, is Class III and IV whitewater that you raft with a guide, a helmet and a safety briefing. They are the same water. They are not the same day out, and an afternoon that starts with somebody saying the rafting is nearby ends badly if nobody checks which river is meant.
For most people staying in a cabin, the Toccoa is the right answer. A float on the upper river takes an easy couple of hours, works with children, works with a cooler, and finishes with everybody sunburnt and pleased with themselves. The river is knee to waist deep over most of a normal summer run, the current does the work, and the scenery is close-in hardwood forest with the occasional gravel bar to stand up on.
The thing that stops people doing it independently is the shuttle. A point-to-point float needs a vehicle at each end, and a family in one rental car cannot solve that problem. This is the whole business model of the tubing outfitters on this river, and paying one of them fifteen or twenty dollars a head is a far better idea than a complicated plan involving two cars, a bicycle and somebody's patience.
Two calendar facts shape the season. The upper Toccoa needs water, and by late August after a dry summer the shoals get bony enough that a two-hour float turns into three hours with dragging. The Ocoee, by contrast, does not depend on rain at all: it runs on scheduled releases from upstream dams, and the rafting calendar is published in advance, which is why an Ocoee trip can be booked for a specific date in March and a Toccoa float cannot be guaranteed for a specific water level in September.
And one hard line. Whatever you do, do not put a tube in below Blue Ridge Dam. That water comes off the bottom of a deep reservoir, it is cold in August, and its level changes when the turbines run. It is superb trout water and a thoroughly bad float.