The quest for an optimal solid-state laser amplifier medium for Inertial Fusion Energy has been investigated through a systematic analysis of more than a hundred optical materials, including glasses, single crystals and ceramics. This work emphasizes the intrinsic difficulty of simultaneously satisfying the key spectroscopic and thermo-mechanical requirements that ultimately govern both the selection of the gain medium and the overall architecture of an IFE driver. As no single material fully satisfies all performance criteria proposed in this paper, material selection must inevitably rely on priority trade-offs driven by appropriate choices between spectroscopic and thermo-mechanical properties as key parameters. Beyond single crystals and glasses, ceramic materials are identified as a promising technological pathway, offering improved thermo-mechanical properties and the capability to fabricate larger-scale components with laser-grade optical quality. Several directions for future research are also identified. In particular, co-doping approaches and composite material designs may provide additional degrees of freedom for optimizing both spectroscopic performance and thermo-mechanical stability.