Most of this thread has been about rhythm. I often do rhythmic accompaniments on an adapted washboard or other percussion, but in several different ways.

There are about three different levels of perceptibility I might aim at. In a large and potentially shambolic group, I may play so the other instrumentalists might hear me but the audience won't: effectively being a conductor. You don't have to be loud and dominant to pull everybody together.

Next level up is playing a sort of contrapuntal line, equal with the melody. If it's a listening situation, this is where you get the greatest opportunity to be creative. You don't want to drown the melody out, ever, but you can add a lot of rhythmic decoration. Good pipe bands and Middle Eastern groups do this.

Above that, you may find when playing for dancing that the dancers are really listening for the pulses the percussionist is giving. Their feet are not moving in time to the tricky cuts and strikes in the tune, or that bit where the guitarist slips in a maj7 chord, they just want to know where the pulses are. So you don't want to care very much if you drown the melody out at times, and you don't want to distract from those pulses with elaborate cross-accents and fills.

As to what beats to play - I find the Middle Eastern or Indian idea of a rhythmic cycle really useful. A lot of the time I will play the same pattern (one or two bars long) throughout one part of a tune, switching to a different pattern for each section, with some sort of signalling hook to indicate when a part is ending (this gives listeners a subtle hint of the tune structure). Pipe band books give written-out descriptions of how to do that. I can usually intuit one that works quite quickly but I don't flatter myself I can think one up straight off first time round on a tune I've never heard before - being somewhat familiar with the tune is essential. Usually I don't do a percussion accompaniment unless I can play the tune on something already.