@YCor, do you mean a different one than the one I cite just above the horizontal line? Maybe I should have added it to the references at the bottom of the post?

Dear Geoff, yes, the proof reduces to the case where $G$ itself is $K$-elementary by Witt–Berman, and pretty much my whole proof deals with that case. If you are willing to look at it and tell me if I am missing shortcuts, I will be happy to send you my proof, which at the moment is only fully written out for $K=\mathbb{R}$.

You are absolutely right, I was too hasty. I will leave this up, so that others don't repeat my mistake.

@Derek Holt: oops, right!

The action you have written down at the end is not a left action.