Sure, you can do studio shooting with a Contax G! Here are some studio pictures I shot in a workshop a few years ago, all using a G2 and a 90mm Sonnar:
As you may be able to see even at this small size, the 90 Sonnar resists flare really well, and the non-SLR finder makes it easier to shoot action because you can see the flashes fire as you watch through the finder. The G2's 1/200 sync speed is a plus as well.
As to why you might have gotten underexposed results, it's hard to say -- lack of second-curtain sync certainly has nothing to do with it. (Second-curtain sync is more of a trick effect that only works with on-camera dedicated flashes; the vast majority of studio pictures are shot with ordinary first-curtain sync.)
You might want to double-check that you had the camera set to the X sync speed, or possibly one speed below it -- some large studio flash systems discharge slowly enough that a high X-sync speed may not pick up all the light.
There's also at least a modest chance that your camera's PC connector or sync circuit isn't working properly. One crude way to test this is to put a sheet of white paper out where it will be lit by the strobes, then remove the lens, open the camera back (no film, of course!) and watch through the back as you fire the shutter with the lens mount facing toward the paper. Hold still and concentrate; it's surprising how much your eye can see even during the very brief flash duration. If the sync circuit is working properly, you'll see the paper as brilliant white; if it's off, you'll see a dingey-looking afterimage of the paper. If in doubt, repeat the test with another camera with known-good flash sync, so you'll know what you're looking for.
Whatever you do, don't let anyone give you any guff about a G being unsuited for studio work! You should be able to get excellent shots with it.