It so happens that the galactic plane cuts the ecliptic very close to 90 degrees and 270 degrees ecliptic longitude at the present epoch, which means that the position of the sun passes through the galactic plane at the solstices.
The relevant galactic node on the ecliptic is at λ = 270.0232 degrees for the J2000.0 equinox, and precession should be making that bigger, rather than smaller. So it seems like the boat has already harmlessly sailed, if we're using the standard astronomical definition of the galactic equator.
Somehow, I suspect we're not ...

Grant Hutchison