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Something I always wonder about when Voyager has these issues is this: how long would it take for a probe launched today to catch up to Voyager? I assume there are better engines now, better capability for payload launch, better fuels, etc. I then wonder how long it would take to get that probe ready to be launched assuming that it was given the green light.

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Most of the Voyager acceleration was done by using gravity assist ("slingshot flybys" -- effectively falling toward a large mass like Jupiter and missing). As far as I know gravity has not improved or become better since the 1970s.

Though calculations about gravity assists have improved and the knowledge about the application as well.

It's a bit more complicated than falling and missing. Getting out of the gravity well takes the same amount of energy as you gain from falling into it, so those two things alone would be pointless.

I'm not an astrophysicist but I think you have to do it in such a way that you "steal" some of the planet's orbital momentum.

This exact question has been tackled by Randall Munroe in his "What If?" section:

https://what-if.xkcd.com/38/

Barely related, but on the same spirit, I recommend Kafka's short story "The city coat of arms" for his perspective on the "there are better engines" line of reasoning:

https://www.babelmatrix.org/works/de/Kafka%2C_Franz-1883/Das...

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