> I am not sure if it could be done with conventional rockets with current technology, without multiple gravity assists.

Yes, it can. New Horizons is escaping the solar system only a little bit slower than the Voyager probes, and it was already on an escape trajectory before its single Jupiter gravity assist.

New Horizons is quite a bit slower and will never overtake the Voyager probes distance wise.

I wonder what hope we have for making probes that will last longer (i.e., “do science even deeper in space”) than Voyager 1, Voyager 2, and New Horizons. Specifically, whether we can make RTGs or other power sources that last longer and/or scientific instruments that operate with lower power (without compromising on quality or quantity of observations). Maybe there just isn’t much science to do beyond the Kuiper Belt (or rather, getting 50 years beyond the Kuiper Belt doesn’t improve science much more than making the same observations from the KB itself due to the vastness of space)?

I'm puzzled by the requirement for an alignment. It seems one could gravity assist in arbitrary directions though it might take longer. But should still end up with a similar energy gain.

The gravity assist works by transferring some of the planet's momentum to the craft, and that momentum has a direction, not just an amount.