Does it make sense to say that objects moving at relativistic velocities appear in space-time cross-section? Let object A move at relativistic velocity relative to a frame O. In 4D space-time (Minkowski diagram) the space view of O at any given moment of its own time is a space-like hypersurface (hyperplane, if O is inertial). What O observes of A at any time is the cross-section of A's world-tube by O's corresponding constant-time hypersurface. 
If so, are there any formal objections to saying that "moving objects are observed in space-time cross-section"? 
Note: By "observed" I mean "described in measurable coordinates", not optical observation by a human observer or a camera.  
 A: 
Let object A move at relativistic velocity relative to a frame O. 

Any effect exists even at smaller nonzero velocities, it's just that the effects are smaller. People have even measured relativistic effects at everyday velocities by using very precise measurements.

In 4D space-time (Minkowski diagram) the space view of O at any given moment of its own time is a space-like hypersurface (hyperplane, if O is inertial). 

The term space view is not standard. Physicists go to great lengths to use the same standard terminology and to mean the same things by the same terms. And if what you mean is that we measure length of objects in a frame by considering events with the same time in that frame, then this is 1) known, 2) obvious, and 3) also true in Galilean relativity. Therefore it is not explanatory for Special relativity.

What O observes of A at any time is the cross-section of A's world-tube by O's corresponding constant-time hypersurface. 

This is very confusing (I know what you mean, but pedagogically it is confusing). Why? Because in the frame you measure it, the events in question are simply simultaneous. Describing them differently is confusing. Saying cross section when you mean simultaneous makes someone think of a tilted hyperplane.
And there is a good way to bring that up. Of you fixed one frame and looked at the events that other frames consider simultaneous you will see a bunch of tilted hyperplane. This is useful when discussing relativity of simultaneity. Which indeed is related to length contraction. But geometrically when you draw that tilted hyperplane if looks like a longer line in the fixed frame, you have to know that those longer lines when tilted have a smaller proper length.
Usually people discuss length contraction, time dilation, and the relativity of simultaneity. Your talk about cross sections might be better suited for a discussion about how these three things are related to each other.
It dies not seem like a good idea of how to introduce or explain the physical cause of length contraction because you already have to know that tilted plane of simultaneity has a small proper length intersection with the object even though when you draw the line segment in the rest frame of the object the line appears to be a greater 4d euclidean distance line segment.
It requires assuming what you are trying to show (length contraction). Or else if you vague up what the plane of simultaneity looks like and how to measure distances between events then it becomes something already true and known.
So it is either the educational equivalent of question begging (assuming what you want to show) or it becomes something so obviously true that it is also true of Galilean relativity (that you measure lengths by measuring the end points at the same time) and so is uninformative.
It is uninformative either way.

If so, are there any formal objections to saying that "moving objects are observed in space-time cross-section"? 

It sounds like you are saying that we measure lengths by measuring things at the same time. This is true but doesn't explain special relativity.
And since it is nonstandard it will sound like you are saying something about visual appearances. But that is nothing next to the fact that you simply aren't conveying content.
People that know relativity (like yourself)/can read the correct things into what you say. People that don't know relativity (people that might read your text work pedagogical purposes) can potentially read the wrong meaning into it, or read nothing into it. So it is not good, unless perhaps in the right context.
edit based on comments
I objected to using ambiguous terminology (cross section) when a standard term exists (hyperplane of simultaneity, or just the simpler "simultaneous"). A cross section generally need not be spacelike, you can have hyperplanes with timelike tangents, spacelike tangents, or even lightlike tangents.
I objected to describing the plane of simultaneity of one frame in another frame without making a big deal about there being two frames and how it is about one frame's judgement of a different frame. This is essential to length contraction because length contraction is about two frames not a thing that happens when things move relative to absolute space. And I know that you know that, so I'm criticising the pedagogy.
And you asked why I bring up length contraction. If you had just one frame you could just discuss the set of simultaneous events and you wouldn't need new nonstandard terminology (you can mention that it is relative if you want) but then the new terminology seems overkill unless you are trying to discuss two different frames.
It is possible that the question here just reads like overkill for saying that spatial measurements are made relative to a plane of simultaneity for some observer. If I made it more complicated than you intended I apologize.
You say moving objects are measured in cross section, that is unnecessarily restrictive because all objects, moving or not are measures in a plane of simultaneity. And it is also too vague since a cross section need not have a timelike tangent. And finally there is a poor history of using nonstandard terminology to describe already know concepts, which isn't a criticism per se, but will affect how something is read.
A: The "different hypersurfaces" effect should be enough to describe the relativity of simultaneity; you will probably not have completely described length contraction or time dilation with your explanation.
As long as you're not trying to completely describe everything but just to describe part of them, then no, there are no formal objections to the observation that your "time-slices" are parallel spacelike hyperplanes in the Minkowski space, and you can speak of your impressions of their spatial coordinates as being a "cross-section of spacetime" in just this way.
A: Your interpretation is formally correct. At least in the special relativity realm.
You can prove this by noting that transformations between systems in relative speed to each other are actual rotations in a space with an extra dimension.
Think like this: if the world is bidimensional, and you have a circle, and let's assume that it lasts for certain time (though this assumption is not needed, I just like it better) the in the 3D world of 2Dspace+time it will be a cylinder. Now if seen by an observer at high relative speed, say in the x direction, if should appear like an ellypse whose x width is shorter and the y width is unchanged. The same results from rotating the 3D cylinder around the y axis! It's section in the plane is this ellipse! And it's dimensions in time (duration) will be longer! Both geometrical effects linked together! Let me know if you want the equations too!
