We can do more then one thing at a time. Alpha Centauri is the closest, easiest star to study by far. The technology to find Earth-like planets is pretty much

We can do more then one thing at a time. Alpha Centauri is the closest, easiest star to study by far. The technology to find Earth-like planets is pretty much an entirely separate field of astronomy to the technology required to characterize those planets (i.e. JWST doing atmospheric-transient spectroscopy). Whatever technique you apply to some "better" candidate would be the same as a technique which works well on Alpha Centauri, so that's the place you use it (absent something more convenient - like a space probe hanging out looking back at Earth with the same methods).