It is entirely possible to do this in databases. That is how it is actually done. The limitations of GEOS are not the limitations of software, it is not a particularly sophisticated implementation (even PostGIS doesn’t use it for the important parts last I checked). To some extent you are affirming that there is a lack of ambition in this part of the market in open source.
I can sort of see your point about the merits of global, spheroidal geometry, certainly from a user's perspective. But there's no getting around the fact that the geometry calculations are both slower (I'm tempted to say "inherently"…) and far more complex to implement (just look at how painful it is to write a performant, accurate r- or r*-tree for spherical coordinates) along every dimension. That's not going to change any time soon, so the projection workflow probably isn't going anywhere.
I'm actually curious, speaking for yourself, what kind of analysis you're doing where something like NAD83, or UTM does not give you enough precision? Is this actually "real world" geospatial data? If I have a soil model, I have a very localized analysis, and if I have a global climate model, we're talking kilometers for grid cells. In all these cases, the collected data has built in geolocation error MUCH grater than most decent projections...
So, what analysis are you doing where you need centimeter precision at global scale of thousands of kilometers? Sounds really interesting. The only time I've seen this, is doing space flight simulations where the error really accumulates into the future.