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I only have surface level knowledge of String Theory, but my understanding is that strings vibrate in simple harmonic motion and that different frequencies correspond to different particles. Since idealized springs are simple harmonic oscillators, you could perhaps say that, in some sense, the strings in String Theory are springs.
But maybe that's what inspired your question. If you're asking why they can't be springs in a more literal, geometric sense, then I would speculate that it's related to the world sheet that a spring would trace out as it propagates through spacetime. A world line describes a trajectory of a point particle not just through space, but through time as well - thereby describing the history of the particle's motion. In quantum field theory, these world lines are used in Feynman diagrams to describe interactions between particles. However, these diagrams always have sharp interaction vertices. In other words, the interaction occurs at a specific point in spacetime, which is problematic in terms of relativity (different observers should not need to agree on when a spacetime event occurred). For reasons I don't understand, this can give rise to infinities (ultraviolet divergences) when doing certain calculations. These have to removed through renormalization, but apparently this doesn't work when trying to develop a quantum theory of gravity.
In the case of a one-dimensional object like a string, instead of tracing out a world line, it traces out a two-dimensional surface called a world sheet. A consequence of this is that the sharp vertices of Feynman diagrams disappear: while an interaction did occur globally, it did not occur at a specific point in spacetime (different observers will see the event occur at different times, so no relativity issues). This eliminates the ultraviolet divergences and the need for renormalization (again, apparently), allowing for a full quantum theory of gravity. If you were to change the geometry of the strings to something more spring-like, my guess is you would no longer get this nice behavior.
Thank you for your amazing answer! This explains my concerns exactly.