We analyze neutral atom Rydberg CZ gates based on adiabatic pulses applied symmetrically to both atoms. Analysis with smooth pulse shapes and Cs atom parameters predicts the gates can create Bell states with fidelity F>0.999 by using adiabatic rapid passage pulses. With globally optimized adiabatic pulse shapes, in a two-photon excitation process, we generate Bell states with fidelity F=0.997. The analysis fully accounts for spontaneous emission from intermediate and Rydberg states, including the Rydberg lifetime in a room-temperature environment, but does not include errors arising from laser noise. The gate protocols do not require individual addressing and are shown to be robust against Doppler shifts due to atomic motion.