tl;dr parent and parent's link are not wrong, just very simple. Unfortunately it does not make explicit that "at least a little" does not mean the entirety of the rotation, or even its dominant component.

In the context above gravitational shear is the first of the empirical https://en.wikipedia.org/wiki/Oort_constants (differential rotation shears the non-solid disc). Differential rotation means gas in the direction of the galactic centre drags a bit compared to gas further from the centre, effectively acting as a small torque on a large gas cloud. If the gas cloud has no net rotation at all, this torque will develop it. That's what I take the sentence you focus on to be trying to say.

Unevenness is related to the Jeans instability, which describes the failure of internal gas pressure to prevent gravitational collapse. Chandrasekhar developed a more complicated treatment useful for when the angular momentum of the bulk gas cloud is non-negligible.

Roughly, collisions within the gas cloud convert kinetic energy to light which carries energy out of the collapsing cloud. The cloud consequently cools and contracts gravitationally.

However, still roughly, superimposing a net rotation on random motions within the gas prevents collapse in the direction perpendicular to the net rotation's spin axis. So contraction is mainly along the spin axis. Result: a disc, which is essentially what parent's link says.

The coupling of the galaxy's spin to the solar system's spin is at best weak; the solar system's spin axis is tilted about 60 degrees from the galactic plane; the plane is perpendicular to the galaxy's spin axis, in line with the previous paragraph.

Here's a diagram <https://i.stack.imgur.com/VIipS.jpg>.

The sun's path around the centre of the galaxy is a bit messy compared to a kinematically hot star (which will feel perturbations like the bar or spiral arms less). The sun tends to bob up and down relative to the midplane of the galactic disc. The Oort constants are local and position-dependent, so the solar system's migration -- and any migration of its precursor -- means wandering into regions with different gravitational shear.

Questions that afaik are still open (but I would be happy for a galaxy or solar system dynamics person to correct me, particularly if the correction includes magnetics and chemodynamics/metallicity-dependent pressure!): has the solar system's spin axis tilt to the galactic disc evolved since formation? Accepting the Coatlicue hypothesis, was the remnant of the explosion of the heavy star that was the precursor of the solar system (and others) aligned with the star's rotation? Was that precursor star's rotational axis aligned with the galactic disc? The remnant was almost certainly not of uniform density. Alternatively accepting the Wolf-Rayet bubble Giant Molecular Cloud (GMC) nebular hypothesis, did the GMC fragmentize under galactic shear, and if so was "our" fragment's spin dominated by that initially, with spin perpendicular to the disc but evolving to the present tilt, or was our fragment's spin axis initially tilted close to the present approx. 60 degrees to the galactic disc where it has since remained?

Finally, also afaik, the spins (more broadly the angular momentum vectors) of stars in the local bubble are essentially random. So repeat the questions in the preceding paragraph for each of those...