Context. The characterization of white dwarf atmospheres is crucial for accurately deriving stellar parameters such as effective temperature, mass, and age. However, the inclusion of physical processes such as convective mixing and convective dilution in current white dwarf atmospheric models offers a prediction of the spectral evolution of these objects. To constrain these models, accurate observational data and analyses are necessary. Aims. We aim to classify the population of white dwarfs up to 500 pc into hydrogen-rich or hydrogen-deficient atmospheres based on Gaia spectra and to derive an accurate spectral type-temperature distribution, namely, the ratio between the number of non-DAs to the total number of white dwarfs as a function of the effective temperature for the largest observed unbiased sample of these objects. Methods. We took advantage of the recent Gaia low-resolution spectra available for 76 657 white dwarfs up to 500 pc. We calculated the synthetic J-PAS narrow-band photometry and fit the spectral energy distribution of each object with up-to-date models for hydrogen-rich and helium-rich white dwarf atmospheres. We estimated the probability for a white dwarf to have a hydrogen-rich atmosphere and validated the results using the Montreal White Dwarf Database. Finally, precise effective temperature values were derived for each object using La Plata evolutionary models. Results. We successfully classified a total of 65 310 white dwarfs (57 155 newly classified objects) into DAs and non-DAs with an accuracy of 94%. An unbiased subsample of nearly 34 000 objects was built, from which we computed a precise spectral distribution spanning an effective temperature range from 5500 to 40 000 K, while accounting for potential selection effects. Conclusions. Some characteristic features of the spectral evolution, such as the deficit of helium-rich stars at T_eff ≈ 35 000 − 40 000 K and in the range of 22 000 ≲ T_eff ≲ 25 000 K, as well as a gradual increase from 18 000 K to T_eff ≈ 7000 K, where the non-DA stars percentage reaches its maximum of 41%, followed by a decrease for cooler temperatures, are statistically significant. These findings will provide precise constraints for the proposed models of spectral evolution.