This is where Dave's perfect little exposition, summed up in one paragraph, tells you all you need to know for a very musical sounding accompaniment.
Really, there's little left to add to what he says .
Since you contradict yourself just two sentences later, I think I don't need to point out that this is an invalid oversimplification.
Seriously, there's never an "all you need to know." While I don't disagree with anything Geoff or David sketched out, it's far from the "whole picture" (certainly wasn't ever meant to be near in the first place). For example, you have to match the accompaniment with the melody. It's tempting to overdouse a simple melody with a complex counter pointed bass line; that's only fun to listen to for a little while, but after that it becomes cumbersome. For simple melodies, it's fine to break the oom-pah rules every once in a while (for example, a rule is "don't use thirds in the bass." True, but every once in a while it doesn't hurt to accompany a transition dominant-tonic at the end of a phrase with the third of the dominant which leads nicely into the root note of the tonic), but it's almost more important to keep the listening experience as a whole in mind, keeping an appropriate balance between the different voices. My suspicion is that most listeners (including myself) are much more confortable with a solidly played tune in which the melody leads and the accompaniment serves than with a complex polyphonic arrangement. But this again is only a crude rule of thumb; there are plenty of examples of very well done complex variations of "simple" melodies.
On complex melodies, on the other hand, you have many more colors and options at your disposition in the first place, so there's many more options to decorate the accompaniment without sacrificing the melody.
Also, bass lines can be employed to add rhythmic variations and emphases to a piece; for example, a piece that has a strong off beat feeling may very well benefit form a pah-OOM type of bass line (or even one that leaves out the basses on 1 and 3 altogether). Another simple way to add an interesting groove could be, for example, to defer the pah by half a beat (of course, this should only be employed for certain tunes; don't try this on reels I would assume?)
These are only two of many points that defy your statement that "all one needs to know" has been written by Geoff and David. It hasn't, and I don't think they ever intended to. Lots can be added, although for a first approximation, they certainly did very good ground work.