The tip of the red giant branch (TRGB) is an important standard candle for determining luminosity distances. Although several 105 small-amplitude red giant stars (SARGs) have been discovered, variability was previously considered irrelevant for the TRGB as a standard candle. Here, we show that all stars near the TRGB are SARGs that follow several period-luminosity sequences, of which sequence A is younger than sequence B as predicted by stellar evolution. We measure apparent TRGB magnitudes, m TRGB, in the Large Magellanic Cloud (LMC) using Sobel filters applied to photometry from the Optical Gravitational Lensing Experiment and the ESA Gaia mission, and we identify several weaknesses in a recent LMC-based TRGB calibration used to measure the Hubble constant. We consider four samples: all red giants (RGs), SARGs, and sequences A and B. The B sequence is best suited for measuring distances to old RG populations, with M F814W,0 = -4.025 +/- 0.014(stat.) +/- 0.033(syst.) mag assuming the LMC's geometric distance. Control of systematics is demonstrated using detailed simulations. Population diversity affects m TRGB at a level exceeding the stated precision: the SARG and A-sequence samples yield 0.039 and 0.085 mag fainter (at 5 sigma significance) m TRGB values, respectively. Ensuring equivalent RG populations is crucial to measuring accurate TRGB distances. Additionally, luminosity function smoothing (similar to 0.02 mag) and edge detection response weighting (as much as -0.06 mag) can further bias TRGB measurements, with the latter introducing a tip-contrast relation. We are optimistic that variable RGs will enable further improvements to the TRGB as a standard candle.