Best version ive seen is to ditch the idea of sending humans entirely , and instead just send drones with dna instructions and proteins. It reduces the costs by many orders of magnitude.
The idea is to have humans born on site on another star system from synthesized genetic information of existing humans from earth. There is a variation on this where frozen eggs and sperms cells are sent , instead of raw proteins. Upon arrival robotics and AI units would use local materials to create a habitable environment. Once the environment is built on site, the gentic information is synthesized (or sperm and eggs are combined to become embryo). Parenting would be the domain of AI with minor input from earth at 4yr intervals due to light lag.
Barring the ethical constraints, this method works because its very easy to accelerate a craft to at least 10 percent of lightspeed if humans arent onboard, requiring sustenance, radiation shielding, etc. this wouldnt require any exotic material for the hull or special thrusting mechanisms. And the acceleration could very easily be achieved using solar sails or photon bombardment from earth. In fact, Von Brauns nuclear-plate propulsion system from the 70s is still the most cost effective and very applicable here particularly for slowing down.
At that rate you could have the essential human ingredients on alpha centauri in slightly more than 45 years. Any random moon or asteroid in the star system would have enough material to build. If you wanted to populate the Galaxy, you would repeat the process for the next star system. But you would start from alpha centauri, instead of earth.