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Guitar strings aren't tune up to or above their elastic limit.
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That is simply not true. There is some plastic deformation when steel strings are tuned up to pitch. This deformation (which is an extension of the hard drawing process used to make music wire) actually makes the steel stronger. Music wire, which is plain carbon steel, gets its strength during the drawing process, which elongates the grains. Where a higher alloy may become brittle, the low alloy used in music wire is more ductile.
Old strings become harder to tune because they are no longer perfectly uniform along their length. Perfect intonation can only be achieved with a string that is uniform. But I don't believe the typical amount of reduction in cross section is enough to feel as lowering the tension.
I believe the main reason for this perceived effect of lowered tension is a reduction in the bending stiffness of the wound strings, due to loosening of the windings. This is one reason that strings go dead.