By introducing a notion of an ideal large-scale filter, a formal statement is given of the hypothesis of the quasi-steady quasi-homogeneous nature of the interaction between the large and small scales in the near-wall part of turbulent flows. This made the derivations easier and more rigorous. A method is proposed to find the optimal large-scale filter by multi-objective optimization, with the first objective being a large correlation between large-scale fluctuations near the wall and in the layer at a certain finite distance from the wall, and the second objective being a small correlation between the small scales in the same layers. The filter was demonstrated to give good results. Within the quasi-steady quasi-homogeneous theory expansions for various quantities were found with respect to the amplitude of the large-scale fluctuations. Including the higher-order terms improved the agreement with numerical data. Interestingly, it turns out that the quasi-steady quasi-homogeneous theory implies a dependence of the mean profile log-law constants on the Reynolds number. The main overall result of the present work is the demonstration of the relevance of the quasi-steady quasi-homogeneous theory for near-wall turbulent flows.