24 pages, 15 figures, first paper of a series of two ; International audience ; The $^{12}$C+$^{12}$C reaction at 95 MeV has been studied through the complete charge identification of its products by means of the GARFIELD+RCo experimental set-up at INFN Laboratori Nazionali di Legnaro (LNL). In this paper, the first of a series of two, a comparison to a dedicated Hauser-Feshbach calculation allows to select a set of dissipative events which corresponds, to a large extent, to the statistical evaporation of highly excited $^{24}$Mg. Information on the isotopic distribution of the evaporation residues in coincidence with their complete evaporation chain is also extracted. The set of data puts strong constraints on the behaviour of the level density of light nuclei above the threshold for particle emission. In particular, a fast increase of the level density parameter with excitation energy is supported by the data. Residual deviations from a statistical behaviour are seen in two specific channels, and tentatively associated with a contamination from direct reactions and/or $α$-clustering effects. These channels are studied in further details in the second paper of the series.