>>If you do any computer simulation that didn't involve high speeds (i.e., classical physics suffices), you would have the same situation, in which you have to decide on one particular coordinate system, and stick to it for simplicity.
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>Sure, but my suggestion is that the one particular coordinate system we use is observer indepedent.

I don't see a big difference, except that more "variables", so to speak, are involved. In the case of classical physics, the observers have to agree on a standard set of coordinates (origin, directions for X, Y, Z). In the case of Special Relativity, they additionally have to agree on a standard velocity.

Sure, it is general knowledge that in the Special Theory of Relativity, the facts of nature are observer dependent. But that doesn't invalidate the Theory in the least. The Theory is strange, but classical physics can not account for facts that are, how shall we say, strange but observed facts. Such as, that the speed of light in a vacuum is constant for any observer. Although this contradicts classical physics, this agrees with observation.