Why can we see not pointlike but extended objects on the sky? Strictly speaking, our past light-cone allows us to observe light emitted by pointlike events situated on a conical two-dimensional surface around us and extending back in time, as is well illustrated in comoving coordinates. But how is it then possible to see an extended large three-dimensional object like a galaxy on the sky at one glance?
 A: Generally speaking, processes which operate on point-like sources can be integrated using calculus to describe the effects that occur from distributed sources.
Also, it can be elucidate to take a step back and consider how an image is formed by a classical lens. While the story isn't quite the same, there are many "how can this happen" which can start by looking at a lens and asking how it happens in a non-relativistic environment first, and then extend it to the relativistic one.
A: As you correctly write, our past light cone has two spatial coordinates. That's the celestial sphere on which we see objects. So already there it's clear that we don't just see point-like objects, but their two-dimensional projections on that celestial sphere.
Putting yourself in the center of spherical coordinates, the celestial sphere makes the two angles, and the time dimension can be thought of as the third radial coordinate. And indeed if an object is in the way, you can in general not look past it.
In addition to this more philosophical aspect, in reality, distances e.g. to galaxies are far compared to the extend of these galaxies themselves. So the light from the "left" and "right" side of the galaxy enter your eyes at basically the same time, to a very, very good approximation. Similarly, if you e.g. use an infrared telescope to look past a dust cloud in a far-away galaxy, you can see the young stars behind it, but given the distances involved, that light was emitted at essentially the same time as well.
So in that sense it is as you write: We can only see the universe in an instant of time. Using your language, the point-like-ness refers to the time slice, not to the angle on the sky.
