The design of the Level-0 endcap muon trigger for the ATLAS experiment at the High-Luminosity LHC (HL-LHC) and the status of the system development are presented. The HL-LHC is planned to start the operation in 2026. The peak luminosity will ultimately reach $\mathcal{L} = 7.5 \times 10^{34}~\rm{cm^{-2}s^{-1}}$. The new Level-0 endcap muon trigger system reconstructs primitive muon candidates using Thin Gap Chamber~(TGC) hits with an improved momentum resolution to suppress the trigger rate, while achieving an efficiency better than the current system. A high-speed serial link between TGC on-detector and off-detector boards with the recovery clock was demonstrated, and the bit error ratio was found to be lower than $4.8 \times 10^{-16}$. The track reconstruction of primitive muon candidates is based on a pattern-matching algorithm using predefined sets of hits corresponding to tracks. An initial test of the algorithm with the Xilinx evaluation kit VCU118 showed a high efficiency with reasonable memory resources. The muon candidate selection with several subdetectors in the inner layers was demonstrated, using Monte Carlo simulation samples produced with HL-LHC conditions. The selection efficiency for a single muon was estimated to be greater than 90\%, a few percent higher than the current system. The rate was evaluated with proton-proton collision data taken with the random trigger overlaid to account for a number of pileup events of 200, which is expected at the HL-LHC. The obtained value for the momentum threshold of $20~\rm{GeV}$, the primary threshold assumed for a single muon trigger, is about $30~\rm{kHz}$. Precise momentum determination by Monitored Drift Tube is expected to further reduce the rate.