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That’s just a standard chemical rocket for the initial push and then a few gravity assists right?

Couldn’t an ion engine with a nuclear reactor providing the electricity accelerate more over that period? I’m genuinely curious, I don’t know the answer.

Even an ion engine needs fuel. We shot out Voyager 1 in 1977. It has traversed 47 years of distance.

There has been the talk of solar sails as well, but gravity assist propulsion is already so much easier to achieve for satellites.

If I thought 17 km/s was even half of the fastest we would ever go, I’d have to give up a lot of dreams for the future. Since I’m not prepared to do so I will stick out my proverbial tongue at this assertion.

Is there any reason that a new generation of ships couldn't leverage the same (or similar, if not better) gravitational pulls that benefited Voyager?

The "Grand Tour" alignment that Voyagers were initially intended to take advantage of only happens once every 175 years (although if the goal is just max speed I don't know if favorable conditions happen more often than that)