Honestly, I don't even see why you want to do skirmish for anything but bayld. A lot of work and RNG bullcrap for a bunch of inferior weapons.
Well, it can drop some nifty money drops, which can more than pay for the cost of entry. GEO scrolls are one of the primary things there that show up, but there are some others. Everyone gets their own pool too, so you don't have to fight over the drops or sell/split.

IMO though, the system would have done better if they scrapped the whole idea of "obsidian fragments" and instead just made this into mweya plasm (and increased the amount given during the run to make it worth going, obviously; something like between 3-6k or so depending on the difficulty of the run + parts you use). In effect, it would make Skirmish become like a mini-Delve, for those that don't want to wrangle an alliance together. The fragments just seem very useless at present; I don't know if that's something they plan on changing later or what, but right now they feel very redundant and I'm sure they are not the reason anyone goes to do skirmish.