I suppose I'm a little contrary but I don't consider the virtual fibering conjecture to really be a big problem in 3-manifold theory. In an earlier era when it might have been an approach to proving geometrization, sure, but nowadays with geometrization a fixture of the landscape, the problem is far less important.
To me a really big problem in 3-manifold theory would beStill quite significant, but no longer vital, and I'd rank it well below these problems:
Find an algorithmic formalism for the Ricci flow (with surgery). i.e. find a combinatorial formalism for curvature on a manifold and the resulting flow. This should be compatible with means for representing surfaces in the 3-manifold so that surgery can be implemented, for example, a formalism using triangulations of the manifold so that it would be compatible with normal surface theory. Likely you would want a suitable notion of Pachner complex to get this formalism off the ground.
Build stronger connections between the geometric perspective on 3-manifolds and other perspectives on 3-manifolds. I would put problems like understanding the properties of the gordian Gordian graph of knots in here. Or the volume conjecture. 4-manifold theory enters the picture here because the question of how geometrization relates to surgery is a big one. Questions like which (rational) homology spheres bound (rational) homology balls, embedding 3-manifolds in 4-manifolds, etc.