Motivated by recent experiments on the quantum magnet K2⁢Co⁢(SeO3)2, we study theoretically the excitation spectrum of the nearest-neighbor triangular XXZ model in the limit of strong easy-axis anisotropy, within the up-up-down (1/3-plateau) phase. We make an expansion in 𝛼=𝐽𝑥⁢𝑦/𝐽𝑧⁢𝑧 instead of 1/𝑆 and calculate the magnon dispersion for any value of the spin 𝑆 at second order in 𝛼, with two important conclusions: (i) the 1/𝑆 expansion converges very slowly for 𝑆=1/2, making spin-wave theory quantitatively inaccurate up to very large orders, and (ii) compared to the linear spin-wave predictions, our magnon dispersion presents a much better agreement with experimental results on K2⁢Co⁢(SeO3)2, for which 𝛼≃0.08.