The $α$-$η$-$κ$-$μ$ is one of the most generalized and flexible channel models having an excellent fit to experimental data from diverse propagation environments. The existing statistical results on the envelope of $α$-$η$-$κ$-$μ$ model contain an infinite series involving regularized hypergeometric function and generalized Laguerre polynomial, prohibiting its widespread application in the performance analysis of wireless systems. In this paper, we employ a novel approach to derive density and distribution functions of the envelope of the $α$-$η$-$κ$-$μ$ fading channel without an infinite series approximation. The derived statistical results are presented using a single Fox's H-function for tractable performance analysis and efficient numerical computations, especially for high-frequency mmWave and terahertz wireless transmissions. To gain insight into the distribution of channel envelope, we develop an asymptotic analysis using a more straightforward Gamma function converging to the exact within a reasonable range of channel parameters. To further substantiate the proposed analysis, we present the exact outage probability and average bit-error-rate (BER) performance of a wireless link subjected to the $α$-$η$-$κ$-$μ$ fading model using a single tri-variate Fox's H-function. We obtain the diversity order of the system by analyzing the outage probability at a high signal-to-noise (SNR) ratio. We use numerical and simulation analysis to demonstrate the significance of the developed statistical results compared with the existing infinite series representation for the envelope of the $α$-$η$-$κ$-$μ$ model.