While the visible matter of the galaxy is disk shaped, there is also the dark matter which is spherically distributed, and the amount of DM closer to the center of the Galaxy than the Earth is amounts to a significant fraction of the mass closer to the center.I count at least 3 differences between the "galactic barycenter plunge" vs "fall-through-the-earth" scenarios:
-the galaxy is more of a disc than a sphere
-the matter within the solar systems radius with respect to the galactic barycenter is not a uniform density
-there is a disproportionately large over-density very close to the center of the galaxy in the form of a supermassive black hole
The SMBH at the center is roughly 4 million solar masses, this is pretty small compared the mass of the galaxy as a whole. As others have already mentioned, its effects would not become prominent until you were almost all the way to the center. In which case, you could approach the speed of light if you were to swing in close enough to it, but that is true for any black hole and not just super-massive ones.