Planes in sci-fi have their own problems, such as anti-grav.
Those aren't planes, and only a fraction of sci-fi planes work through anti-grav. I wasn't talking about sci-fi planes in the sense of 'look, anti-grav planes!', but rather planes as in the 'wow, sub-orbital plane-travel is commonplace!' kind of fashion.
I would extend that to include things that are inconsistent with the rest of the technology within the story's universe. It doesn't make sense to have a society that has advanced in one field--such as medical nanotechnology--far more than it has in any other, unless there is some important reason why. The author needs to give at least some consideration to the progression from present day technology to the technology of the story setting.
This makes the flawed assumption that we in real life have a roughly equal level of development across all fields, which is most certainly not the case; it just looks to us when we change the distribution around for a story because the distribution of technological progress that we have in reality seems 'normal'. In reality, the distribution of technological progress is far from equal across fields. Furthermore, the way in which we actually use all our technological progress is far from uniform as well: consider the fact that we live in a world covered in a digital communications network that allows for instant communication across the planet; a world in which brain-machine interfaces already exist, and in which multiple nationstates are currently racing to see who can build the first artificial human brain and break the exaflop scale in supercomputer; a world in which we've landed probes on comets; a world in which we can print organs and create lithium-ion batteries the size of a grain of sand; yet somehow the sight of someone using a pencil doesn't strike us as hopelessly anachronistic.
Like William Gibson said; the future is already here, it just isn't very evenly distributed.
None of this is to say that you aren't right to say that the author shouldn't give some thought to why future society x is way more advanced in some fields than others, but it's not quite so remarkable a thing as to always demand an explanation. Maybe if all other technology stood still.
There could be many reasons, but the setting and plot of the story will limit which of those can be used. One probably can't write aliens into the story solely to solve the stasis problem and then not mention the aliens beyond that.
I don't see why one couldn't, actually. Plenty of good stories have done exactly that; so long as you write a good character driven story, it doesn't matter if the seemingly massive plot element that kickstarted it all isn't resolved. In fact, it's often better if it's deliberately left unresolved.
Sometimes the science is the point of the story. Or rather, speculation about the application of the science is the point of the story, and is what makes the story interesting.
Most sci-fi authors would probably disagree. All good science-fiction stories are character driven. The future tech/science is almost always just the means to an end; the way to set up the story rather than the point itself. As you said, the point isn't really the science itself, but the application of said science. Even that, I would say isn't really the point; but rather it's an excuse to explore facets of humanity and philosophy given form by placing characters in a world different from out own, no matter how important the science is in a story, it's still ultimately up to the characters to give it meaning and context. This is true even in most hard sci-fi; the science may take more of a central role in the story, but ultimately the story still stands or falls with its characters. You can have good science-fiction without well written science; but not without well written characters. The best sci-fi of course, mixes well-written characters with intriguing scientific ideas.