That may be true, but it's worth noting that the length of a "generation" of stars varies wildly depending on the type of star. Stars like our Sun might only live 10 billion years or so, but red dwarfs live 100 billion years or more so we could presumably live around them long after all the Sun-like stars have burned out.

Assuming we possesed that capability it seems like it would be simplet to build a dyson sphere around another star and setup shop on a nearby planet.

Stars fail when the elements in their cores stop releasing net energy when fused. This crossover occurs at iron. Adding more mass will just make it's death more violent.

Also the sun contains 99% of the mass of the solar system. The remaining 1% includes Jupiter. I'm not sure how much effect we can have when only leveraging 1% of 1% of the solar systems mass.

Proper management of star lifetime actually requires the opposite approach: subtract mass from the star, so that the star "core" where actual fusion happens can steadily mix with a larger fraction of the remaining mass. The harvested mass can then be stored away as gas-giant planets, and used for miscellaneous purposes.