Earlier quoted context omitted.
The scenario where a gravity assist doesn't work is if turning sharply enough would require your minimum planetcentric approach distance to be less than the radius of the planet - you'd crash into it instead. This is why Voyager 2 couldn't also do Pluto - it would have needed to change course by roughly 90º at Neptune, which would have required going closer to the center of Neptune than Neptune's own radius. The most…
Could Voyager have reduced velocity, glanced off Neptune, waited for the return path on a narrow elliptical orbit, and then boosted to effectively make the 90° turn to Pluto at that time? Was that impossible given its Neptune approach trajectory, or would glancing off and waiting have been more fuel? And, why didn't this vortical model that includes the forward velocity of the sun make a difference for Voyager's orbi…
It also likely wasn't possible to get to Pluto on some future Pluto orbital pass. The limiting factor is likely that Voyager's incoming trajectory to Neptune was already too far beyond solar escape velocity to get into that narrow elliptical orbit you propose. (You'd have to slingshot so close to Neptune's center that you'd hit the planet instead.)
Designing from the beginning to come in slower to Neptune and adjust to encounter Pluto on some future Pluto orbital pass was probably possible, but yeah you might be talking about time scales of Pluto's entire orbit or even multiples of that. (We do similar things for inner solar system missions, like several encounters with Venus separated by multiple Venus-years, but that's on the order of single-digit years and not hundreds.)
The common answer to a lot of these outlandish slingshot questions is usually, yes it's eventually possible by orbital mechanics, but it gets so complicated and lengthy that you may as well just build another separate spacecraft instead. We talk about Voyager's grand tour alignment because it's captivating, but realistically if that hadn't happened we would have just done separate Jupiter-Uranus and Jupiter-Neptune missions instead.
The sun's motion relative to the galaxy doesn't matter for any of this - nothing else in the galaxy is remotely close enough to affect anything, the nearest star is still over 1000x Voyager's distance.