It took years after I learned the Kalman Filter as a student, until I actually intuitively understood the update of the covariances. Most learning sources (including the OP) just mechanically go through the computations of the a-posterior covariance, but don't bother with an intuition other than "this is the result of multiplying two gaussians", if anything at all.

I wrote down a note for myself where I work this out, if anyone is interested: https://postbits.de/kalman-measurement-update.html