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The problem is that in physics those components are very, very important; the coordinate-free approach discards pieces of information that are crucial to the physical realization of these things.
Analogously, suppose you have an object and you want to know how it falls in an atmosphere. Suppose I tell you that it looks like a sphere, once you remove it from 3-dimensional Euclidean space. That's good and all, but there are lots of things that look like spheres once you remove them from 3-dimensional Euclidean space, and some of them will drop like boulders and others will float like parachutes, because it is precisely their embeddings in 3-dimensional Euclidean space that determines how they fall.
In the coordinate-free approach to tensors, the symmetries and such that make everything nice are reached almost precisely by ignoring all the stuff that physicists look at. If you take results from the component-based version and remove the indices, then you get coordinate-free results but the physics disappears, and if you take results from the coordinate-free version and insert indices, all the niceness and smooth geometric results disappear. So having one doesn't really help with the other.

No problem. There just have been a lot of issues with posting in guilds recently, so I wasn't sure.

It's a standard issue in mathematics. All the nice things in mathematics come from using the available symmetries. Even algebraic geometry, in which you assign the notion of coordinates to things, as soon as you can you ditch the geometric object itself and look at everything modulo the local symmetries.
But physics needs numbers. Space and time have coordinates, and if you average over the symmetries, you remove your space and you remove your time (this is an actual, humongous problem in quantum gravity). All those nice simple things aren't nice in coordinates, because coordinates aren't symmetrical by design.
Tensors are a very clear case of this, because a coordinate-free tensor is defined as an equivalence class, i.e. it's defined as a set of things that are taken to one another by some symmetry, and assumes them to be the "same" tensor. All the nice results come from being able to say "well, X isn't actually Y, but it's in the same equivalence class as Y, so things I can say about X I can say about Y".
In the component-based version, it becomes "well, X isn't actually Y, but it's equivalent to Y under some transformation Z, and so things I can say about X I can say about Y as long as I've kept track of how Z affects what I'm trying to say", wherein "how Z affects what I'm trying to say" is a thirty-line computation.

Yes, it refers to Electrical Engineering. I blame a friend for getting me interested in the subject in high school.

I shall never forgive him for it.

@Tensor Calc: Alright then. So, if I care more about physics, I should just pick up a book on GR and learn it through that?

Alrighty then.

The fundamental problem is that a coordinate system will attempt to distinguish between points, and the coordinate-free version will only work if you treat certain points as being the same.
Consider a horizontal line: you can choose to distinguish the points on the line, or you can treat the line as a whole. In the first case you can tell where on the line you are, and you can tell if someone shifts the line to one side, for instance. In the second case, sliding the line to the left or right gives you the same line, so you can't tell if the line has been shifted; this is what I mean by a symmetry: you can do something to the object and not be able to tell.
Certain results in the coordinate-free case only work if shifting to the left or to the right gives you the same line, so any coordinate system that labels the points would immediately break these results, because then the line shifted one unit to the right would be distinguishable, via the coordinates, from the line shifted one unit to the left.
So no, you can't assign "symmetrical" coordinates, since coordinates provide exactly the kind of information that you're trying to ignore.
There are certain notions of "homogeneous" coordinates that don't entirely undo the symmetries, but these are even more difficult to work with than regular coordinates.

I'm speaking of a theoretically infinite line; tensors don't exist in the real world either (and a coordinate system on tensors would not look anything like coordinates of space or time) so trying to restrict our attention to the real world only obscures things (again, the attempt to relate things to what laymen know only causes falsehoods and confusions). Don't think of a line in the real world; lines don't exist in the real world. Think of a mathematical line; it extends forever in both directions, every point on it looks like every other point, and so the only way to tell if it has been shifted along itself is by assigning coordinates to distinguish between points.

Homogeneous coordinates usually refer to what is called a projective space. Consider R³, stretching off into infinity in all directions. We choose some point to be the origin, and we assign a set of rectilinear coordinates to each point, i.e. we choose an x-axis, a y-axis, a z-axis, and a unit length and assign coordinates based on that. Now every point can be described by a triplet of numbers (x, y, z).
Now consider a line L that goes through the origin. If there is a point on L has coordinates (a, b, c), then the point with coordinates (2a, 2b, 2c) is also on L, as is the point (a/2, b/2, c/2), and in fact, any point with coordinates of the form (m*a, m*b, m*c) is on L.
Now we create a projective space by considering each line through the origin as a single object, effectively saying that if (a, b, c) is not the origin, then for any real number m that isn't 0, (a, b, c) is equivalent to (m*a, m*b, m*c). So the x-axis becomes a single point, and the y-axis becomes a single point, and the z-axis becomes a single point, and the line (y = x, z = 0) becomes a single point, and so on.
This is known as the real projective plane, written RP².
How do we assign coordinates to this space? It's a two-dimensional object, so naively we can assign to each point a pair of numbers, right? Well, yes, but not in a way that really helps, since in order to get all the points in a sensible fashion we'd have to let the coordinates become infinity.
So instead, we go back to the coordinates on the original R³. For a given line, L, we take a point on L and consider its coordinates (a, b, c). Then we write that L has coordinates [a:b:c], where the square brackets and the colons let us know that these are homogeneous coordinates, not regular coordinates. So now we have coordinates for each point in RP².
But it's not so easy. If (a, b, c) is on L, then (2a, 2b, 2c) is also on L. So perhaps the coordinates for L should be [2a:2b:2c] instead. Or maybe [3a:3b:3c], since (3a, 3b, 3c) is also on L. How will we choose?
The short answer is that we can't. If we say that L can only be written as [a:b:c] and not [2a:2b:2c] and so on, we immediately leave RP² and end up back in R³. This is bad, because in many cases RP² is the important object and R³ is annoying baggage. This means that in order to have a sensible coordinate system on RP², we need to have a system in which a single point has multiple (in fact, an infinite number of) different labels.
Luckily for us, the coordinate system is not entirely out of control, in that while the individual numbers [a:b:c] are not unique for a given L, their ratios are, even if that ratio is sometimes infinite or undefined (for example, [0:0:c] describes the point that was the z-axis), and each L has a different set of ratios. So any triplet [a:b:c] defines a unique line, and we can tell when two triplets define the same line.
This is a homogeneous coordinate system, in that the coordinates respond to the symmetries of the lines; a line through the origin is symmetrical under uniform scaling, so the coordinates are as well.

Tensor coordinates end up slightly differently, since the problem isn't so much the coordinates themselves as it is keeping track of them. The symmetries in question are also a lot less geometric, but you still have the problem of trying to distinguish between things with multiple labelings. In a coordinate-free version, you simply dispense with the labeling altogether and look at more abstract structures, but then you've lost your labels and can't distinguish individual tensors at all.