This paper presents a performance analysis of two distinct techniques for antenna selection and precoding in downlink multi-user massive multiple-input single-output systems with limited dynamic range power amplifiers. Both techniques are derived from the original formulation of the regularized-zero forcing precoder, designed as the solution to minimizing a regularized distortion. Based on this, the first technique, called the $\ell_1$-norm precoder, adopts an $\ell_1$-norm regularization term to encourage sparse solutions, thereby enabling antenna selection. The second technique, termed the thresholded $\ell_1$-norm precoder, involves post-processing the precoder solution obtained from the first method by applying an entry-wise thresholding operation. This work conducts a precise performance analysis to compare these two techniques. The analysis leverages the Gaussian min-max theorem which is effective for examining the asymptotic behavior of optimization problems without explicit solutions. While the analysis of the $\ell_1$-norm precoder follows the conventional Gaussian min-max theorem framework, understanding the thresholded $\ell_1$-norm precoder is more complex due to the non-linear behavior introduced by the thresholding operation. To address this complexity, we develop a novel Gaussian min-max theorem tailored to these scenarios. We provide precise asymptotic behavior analysis of the precoders, focusing on metrics such as received signal-to-noise and distortion ratio and bit error rate. Our analysis demonstrates that the thresholded $\ell_1$-norm precoder can offer superior performance when the threshold parameter is carefully selected. Simulations confirm that the asymptotic results are accurate for systems equipped with hundreds of antennas at the base station, serving dozens of user terminals.