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I have reviewed [1] and [2] but it isn't clear to me how they prove that space isn't quantized at the Planck length.

Could you explain how these experiments disprove that?

Here's [1] a pop science explanation of that result (and there have been several more in the same vein since then).

A key quote from the article "And given that some quantum gravity frameworks predict that effects should be showing up at that point, perhaps those models are simply wrong, and there is no changing speed of light."

AFAIK, later experiments have even lowered the threshold, making LQG less and less likely, in the same way the LHC results have pushed back supersymmetry arguments by invalidating some versions predictions.

[1] https://www.symmetrymagazine.org/breaking/2009/10/28/gamma-r...

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