Problems with the availability of raw materials on the Polish market have forced wood industry producers to search for new, previously unused species of wood that meet the functional requirements of the target products. Therefore, it is necessary to conduct research on products whose structure is different from the popular offerings on the market. The goal of this study was to analyze the influence of the thickness of Scots pine veneers, also with Alder interlayer variants, on Young’s modulus and stiffness values of plywood-like composites dedicated to flooring applications regarding statistically based quality control of the products. The variables in this research are the thickness of the veneer, which creates the structure of the base layer of flooring materials, and the kind of wood used. This research looked at basic mechanical properties determining the suitability of flooring materials: modulus of elasticity and stiffness. Because both these parameters describe the product quality, the analyses were based on the normal distribution (containing kurtosis) and the creation of Shewart Control Charts for each parameter. Analyses of control charts provide information on whether the projected production process is stable and is able to give predictable results. In turn, the analysis of the kurtosis value allows us to determine whether Young’s modulus and stiffness values obtained for the products are as close as possible to the values assumed by the manufacturers. The thickness of veneers in the base layer of flooring composites can be enlarged, allowing production to be simplified and more environmentally friendly. New types of layered composites for flooring, manufactured by rotary cutting, without the need for quality assessment, with a minimum number of layers, and additionally verified with Shewart Control Charts, may be applied to production. Presented studies show that veneers of different quality classes, having plywood-like structures, can be used for flooring materials and that the thickness of the veneers in the base layer can be increased. In this way, wood can be used without the need for quality classification and with fewer production processes.