Voyager 1 and 2 had to happen in a narrow window of time to take advantage of favorable planetary alignments. If we launched something like this now, it would take an order of magnitude more fuel and still not be able to go as fast. The next window for exploring interstellar space so efficiently isn't until 2150.
Of course there are other long-term missions we could be trying now; but it's not quite so straightforward (due to the mechanics of space-time) as we're led to believe.
For that matter, we could consider this whole anthropocene climate change thing we're doing to be one long-term experiment. You're welcome, children of the future, for all this wonderful data we're generating for you.
What we use the SpaceX starship - fully refueled in orbit - as a second stage? Put a 3rd stage in/on it to launch a probe. Fire the 3rd stage down in jupiters gravity well to get the most from it, or whatever trickery can be had. I suspect we could get to the heliopause a bit quicker even without such nice planetary alignment.
With SpaceX pushing down launch costs, it would really be possible to launch these probes regularly and in large nunmbers. Maybe there could be an NGO that collects donations and builds thousands of always the same exploration probes, to reduce cost. If each proble was named by whoever donated the money for it, that would probably increase donations further.