We did “burn through synonyms”, though: it’s just that “gay” and “queer” are two synonyms that happen to have stuck around up to now. Other synonyms that were once considered acceptable for descriptive use, such as “fairy”, “sodomite”, and “pansy”, have been “burned through” and thrown away.
We “burn through synonyms” in this way all the time, even when the word in question, like “thug”, starts out as a specific descriptor that isn’t exclusive to any one race/gender/sexual orientation. If it gets attached to a marginalized group, it often ends up acquiring connotations of bias and subsequently discarded.
For instance, the word “mammy” used to mean just a child’s nurse/nanny, with no racial category implied. Once it became identified with uneducated-motherly-black-woman-devoting-herself-to-white-employer’s-children, everybody stopped applying it to non-black nannies. And eventually the racial bias inherent in the connotations of “jolly domestic nurturing black woman” resulted in the term falling out of use almost entirely.
Does it bother you that we introduced other terms to replace “mammy”? I doubt it. That’s just the cost of doing business with language and the endless subtle cultural implications of language.
Another word that may be heading for the burn pile is “illegal” used as a noun, especially in the plural form “illegals”. Of course, it has a perfectly good non-racist core meaning as a short form of “illegal immigrant”, in the sense of an undocumented or unauthorized resident. And there are undocumented immigrants of all races all over the world that could with equal validity be called “illegals” in that sense.
But many non-Latino Americans use “illegals” to speak of specifically Latino undocumented immigrants. And it’s not uncommon for racist non-Latinos in the US to refer to groups of Latinos as “illegals” even if they have no idea what their residency status happens to be. The term “illegals” in that usage isn’t referring to faked papers: it’s referring to speaking Spanish or playing salsa music or shopping at a bodega or other characteristics related to culture rather than legality.
Illegals: Undocumented immigrants -> Latino undocumented immigrants -> Latinos who seem “unassimilated/foreign” irrespective of whether they’re actually in this country illegally.
Thugs: Violent brutal guys -> violent brutal black guys -> black guys who seem “intimidating” irrespective of whether they’re actually guilty of any violence or brutality.
If we end up losing both those terms from common usage because they’ve become more about expressing racial prejudice than about identifying a particular type of behavior that’s not racially specific, well, so be it. Other synonyms to describe those particular types of non-racially-specific behavior will be found.