Okay, if we travel at 20 km/s, how long does it take to travel one-light year distance? My answer is 15,000 earth years. How long do you want to take to reach *nearby stars*? Remember Newton's 2nd law of motion here too.
How fast can you accelerate a probe using gravitational assists? I do not know how to do the math, but if the elusive planet 9 turns out to be a primordial black hole as some have speculated, then you should be able to get one heck of a gravitational assist from it.
I would think that it would be possible to obtain a velocity of 200 to 300 km per second just using a space rated nuclear fission drive. We are close to the ability to build such a space power plant now. By the time we are able to harvest sufficient liquid hydrogen from, say, Ceres, we ought to be able to build such a craft. But realistically we would need to achieve a velocity of 30,000 km per second (roughly ten percent of the speed of light if I did the numbers right) before it would be worthwhile to bother with. I do not have any opinion on whether or not that kind of velocity would be achievable with a fusion drive, which is still very far in the future. We would probably being selling liquid hydrogen like we do a tank of gas in the asteroid belt before that happened.
I go back to my earlier comment using one-light year distance and 20 km/s velocity model. If we travel at 300 km/s, it will take nearly 1,000 earth years to travel one-light year distance from earth. It looks like from this discussion, much work remains to travel to the stars (and beyond)
Of course another big problem with sending a probe a couple of light years distance to the nearest stars is how would we send back any data that the probe acquires at that distance. Based on existing radio or laser communication technology you'd need a pretty powerful radio or laser transmitter on the probe to send back any data, which it seems to me would rule out the recently proposed concept of laser propelled probes with a mass of about 1 gram! https://www.space.com/29950-lasers-power-tiny-interstellar-spacecraft.html