We propose a new analysis of small-scale cosmic microwave background (CMB) data by introducing the cosmological dependency of the foreground signals, focussing first on the thermal Sunyaev-Zel’dovich (tSZ) power spectrum, derived from the halo model. We analyse the latest observations by the South Pole Telescope (SPT) of the high-ℓ power (cross) spectra at 95, 150, and 220 GHz, as the sum of CMB and tSZ signals, both depending on cosmological parameters and remaining contaminants. In order to perform faster analyses, we propose a new tSZ modelling based on machine learning algorithms (namely Random Forest). We show that the additional information contained in the tSZ power spectrum tightens constraints on cosmological and tSZ scaling relation parameters. We combined for the first time the Planck tSZ data with SPT high-ℓ to derive new constraints. Finally, we show how the amplitude of the remaining kinetic SZ power spectrum varies depending on the assumptions made on both tSZ and cosmological parameters. These results show the importance of a thorough modelling of foregrounds in the cosmological analysis of small-scale CMB data. Reliable constraints on cosmological parameters can only be achieved once other significant foregrounds, such as the kinetic SZ and the cosmic infrared background (CIB), are also properly accounted for