Well voyager depended on a solar system alignment that only happens every 175 years(?) so it'd be a while before we get that same advantage again. The longer it takes the further of a head start voyager gets?
The headstart doesn't really matter, anything faster than Voyager will catch up eventually
Starship could be refueled in orbit. That should then be able to reach those kind of velocities with enough capacity to even include a small 3rd stage inside with the payload.
That alignment is only necessary to do the Grand Tour, to visit all four outer planets in one mission. Voyager 1 actually didn't do the Grand Tour, it only visited Jupiter and Saturn, you're thinking of Voyager 2. This alignment is also not even necessary to attain the highest speed, Voyager 1 is even faster than Voyager 2.
A flyby of both Jupiter and Saturn can be done every two decades or so (the synodic period is 19.6 years)
https://en.wikipedia.org/wiki/Grand_Tour_program