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TODO. I think this is the key point. Notably, symmetry implies charge conservation.
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More precisely, each generator of the corresponding Lie algebra leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.
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This is basically the local symmetry version of Noether's theorem.
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Then to maintain charge conservation, we have to maintain local symmetry, which in turn means we have to add a gauge field as shown at Video 93. "Deriving the QED Lagrangian by Dietterich Labs (2018)".
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Forces can then be seen as kind of a side effect of this.
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