Its not unrealistic sci-fi. We have particle accelerators that operate at much, much higher energies. The kind of energy that would succesfully leave earth and our sun behind if these particles were shot out of the LHC. That is to say, you'd be right that any vessel carrying fuel wouldn't reach such speeds. And while its true that even the smallest sattelites today are still at least 1000grams. Getting that up to relativistic speeds would require on the order of 2 billion MJ of energy. Or the equivalent of a large power plant runnning for 20 days to supply all the energy.
But that's only the start, any advancements in making sattelites on a chip could massively reduce the weight and therefore power requirements. Even with all losses accounted for. I believe this could be doable in the next 50 years. You just need to build the most powerful laser anyone has every built and power it for as long as you can.
You’ve shifted and redefined the goalposts and even then requires orders of magnitude advancement in multiple areas (power generation, lasers, satellite shrinkage etc) and doing it all in space. Thinking any of this happens in a lifetime of anyone alive today seems unrealistically optimistic. And all of this ignores something critical OP said:
> we'll expect people to live hundreds of years by the time we can send a ship to alpha centauri.
Even hundreds of years is not a realistic life expectancy to reach the nearest star and requires not just an enormous overabundance of energy, the fuel source has to be available at the midway point so that you can decelerate. Humans at another star without discovery of faster than light travel mechanisms (mainly wormholes) seems purely in the realm of science fiction.