" Doppler shift, which involves monitoring a star’s brightness levels over time"
Doppler shift, which these guys used, is using a spectrometer to look for a periodic shifting of spectral lines of the star from the planet pulling the star to and away from earth as the planet orbits it's star. Planets are really small compared to a star. So these shifts are small for even big planets. One of the not fun problems with doing this is earth's orbit adds it's own shift into that spectrometer data.
The monitoring a star’s brightness levels over time is different. Its problems are that planets are really small compared to a star, so the brightness change is small to very small. And also you have to make sure you're not counting sun/star spots as planet transits.