The light output of deep ultraviolet (UV-C) AlGaN light-emitting diodes (LEDs) is limited due to their poor light extraction efficiency (LEE). To improve the LEE of AlGaN LEDs, we developed a fabrication technology to process AlGaN LEDs grown on SiC into thin-film flip-chip LEDs (TFFC LEDs) with high LEE. This process transfers the AlGaN LED epi onto a new substrate by wafer-to-wafer bonding, and by removing the absorbing SiC substrate with a highly selective SF6 plasma etch that stops at the AlN buffer layer. We optimized the inductively coupled plasma (ICP) SF6 etch parameters to develop a substrate-removal process with high reliability and precise epitaxial control, without creating micromasking defects or degrading the health of the plasma etching system. The SiC etch rate by SF6 plasma was ~46 \mu m/hr at a high RF bias (400 W), and ~7 \mu m/hr at a low RF bias (49 W) with very high etch selectivity between SiC and AlN. The high SF6 etch selectivity between SiC and AlN was essential for removing the SiC substrate and exposing a pristine, smooth AlN surface. We demonstrated the epi-transfer process by fabricating high light extraction TFFC LEDs from AlGaN LEDs grown on SiC. To further enhance the light extraction, the exposed N-face AlN was anisotropically etched in dilute KOH. The LEE of the AlGaN LED improved by ~3X after KOH roughening at room temperature. This AlGaN TFFC LED process establishes a viable path to high external quantum efficiency (EQE) and power conversion efficiency (PCE) UV-C LEDs.