I'm with Krista on this one. I've cleared Garuda ex 4 or 5 times now, and every time it was using Sup only method. The key is simply to have a good balance between melee and ranged so that DPS doesn't have to stop when the melee have to get clear of WW.
As monk I now run with the MT during tornado phases, quickly get two rotations and DOT's in, wait for brown wind, run through tornado to the ranged and healers. There I wait for WW, and as soon as it's gone I use Shoulder Tackle and either help finish off Sup if the ranged haven't done so already, or DPS and DOT Garuda if Sup is already dead. Once I jump back on Garuda, I'd try and get two rotations with DOT's in, wait for brown wind, then run back to the healers again to avoid Garuda's second WW and to prepare for DPS-ing Spiny for when she jumps.