No, Mexico did not do so out of making everyone equal, but did so as a way to screen existing caste and colorisms that are practiced currently within Mexico, both in media, politics, lifestyle and general discourse. It is an attempt to erase the existing lines and divisions that were the legacy of colonialism and henceforth have been attempted to be forgotten and hidden under the veneer of 'Mestizo nationalism'. When the legacy of colonialism is not addressed readily and is instead hidden, it does not point to remedies of the lived realities of the people in question; rather it benefits the ruling European class, who appease those with grievances whilst retaining their traditional division from the more 'Indian looks' and the traditional role of privilege and power in institutions of power, hence racism. The term Latino is an example of this system, which denies the indigenous and near relations of indigenous and the African diaspora in Latin America and identity and places them under an umbrella of what has been termed, 'white supremacy' and 'Eurosupremacy' within countries of the Western Hemisphere. So, when you say, so and so is a Euro Latino or a brown Latino, this is playing into that existing system which is denying the existence, grievances and the legacy of colonialism within these lands. Rather, the existence of European features within Mexico, is simply a result of colonialism and conquest.
I do not know of this idea of European looking peoples speaking indigenous tongues as their native tongue. Even so, there are many Boers who speak Xhosa, this does not mean that there was/is not a sense of colorism that prevails in general societal discourse within South Africa, the same is true for Mexico.
Well part of the issue with the counting of Mestizo, is from what I am aware of, the system Spain implemented was a system of varying degrees of admixture that descended and ascended. Mestizo, traditionally referred to a seamless 50-50 ratio or roundabout mixture between Indigenous and European populaces. However, the system also counted many categories of 60% Indigenous, 75% Indigenous, and so forth and likewise, mixtures called 'Zambo' between Indigenous and African populaces (which do exist in Mexico, African minorities tend to be one of the most erased groups in the country of Mexico). The Mexican government intentionally assumes everyone is mixed and Mestizo, despite the colonial system denying this extrapolation and likewise, the general society of Mexico denying this system and formulation through its interactions of colorism and its bias toward the people of European ancestry or complete European features in positions of power/privilege.
There are many cases of Anglophobia in the US in the past, even from supposed English descent. Do we assume thus, that due to their dislike of the British Empire and its methods, that the European US peoples had some sort of solidarity with its existing populaces who were of non-European ancestry or of generally less than total European ancestry (see Louisiana)? Nay, rather the US displayed and still displays a similar custom and system as that of Mexico, in regards to the legacy of colonialism and colorism, whilst claiming equality and so forth.