I like this question because it reminds us that the future might be easier than the present. I find shells generally a pain that will hopefully go away one-day, or at least be relegated to special applications which are really pushing the boundaries. I see it as more a question of "is my software powerful enough that I can use solids?" not "are solids or shells better for this task?". Most of the limitations of solids are really just limitations in the software which don't always exist, or need to exist. Let me go through some:
Solid meshes are too big - That's what happens with a cheap tet automesher. A single 20 node hex element can often do as well as an 8 node shell even when it's thin and subject to bending. Both have about the same number of DOFs. If you do need more elements through the thickness, 2 or 3 is still only a factor of 2 or 3 more, not huge. Not the difference between minutes and weeks. That does depend somewhat on how thin it is though.
Solid elements can't properly transfer bending moments - Yes they can, they just look more like force couples instead. They go beyond shells and also properly transfer out-of-plane loads that you might not have realized were significant.
Solid elements can't report bending moments and forces - That's a limitation of software, not the elements. It's just harder for the programmer so often they don't do that. You might also see it as a limitation of design codes that ask you to find the more human-friendly and artificial forces and moments instead of the more natural general stress state.
Shells are meant for thin parts - Yes, and typewriters are meant for writing but you wouldn't use a typewriter when you want to write something. Even for thin parts, solid elements are more correct than shells and beams which have extra subtle simplifying assumptions that make them harder to use safely. I've seen people incorrectly using shells and beams because their parts were thin or beam-shaped, but they didn't realize that stresses at the details were wrong or their beams aren't nearly slender enough for the no-shear-stiffness assumption of simple Euler beam elements.
You can't easily change the thickness of solids - With FEA integrated into CAD, and fast meshing there's little or no advantage over shells here. Shells can actually be harder to change the thickness of because that might involve moving the midplane too.
You can't use some material models with solids - Again that's just a limitation of software.
If you don't believe me, have a look at Calculix. It internally converts shells and beams to solids for solving, then converts them back again for output!
FEA software feature chart