A New Catalog of 100,000 Variable \emph{TESS} A-F Stars Reveals a Correlation Between $δ$ Scuti Pulsator Fraction and Stellar Rotation

δ Scuti variables are found at the intersection of the classical instability strip and the main sequence on the Hertzsprung-Russell diagram. With space-based photometry providing millions of light-curves of A-F type stars, we can now probe the occurrence rate of δ Scuti pulsations in detail. Using 30-min cadence light-curves from NASA's Transiting Exoplanet Survey Satellite's (TESS) first 26 sectors, we identify variability in 103,810 stars within 5-24 cycles per day down to a magnitude of $T=11.25$. We fit the period-luminosity relation of the fundamental radial mode for δ Scuti stars in the Gaia $G$-band, allowing us to distinguish classical pulsators from contaminants for a subset of 39,367 stars. Out of this subset, over 15,918 are found on or above the expected period-luminosity relation. We derive an empirical red edge to the classical instability strip using Gaia photometry. The center where pulsator fraction peaks at 50-70%, combined with the red edge, agree well with previous work in the Kepler field. While many variable sources are found below the period-luminosity relation, over 85% of sources inside of the classical instability strip derived in this work are consistent with being δ Scuti stars. The remaining 15% of variables within the instability strip are likely hybrid or γ Doradus pulsators. Finally, we discover strong evidence for a correlation between pulsator fraction and spectral line broadening from the Radial Velocity Spectrometer (RVS) aboard the Gaia spacecraft, confirming that rotation has a role in driving pulsations in δ Scuti stars.