When theorists talk about Scarlatti laying the foundation of harmony, partimento is probably the foundation they are referring to. Functional harmony of the 19th century is just a more powerful abstraction.
Second, while they did think differently about harmony, you can describe it in modern terms. A description like Cmaj7 doesn't mean anything else than a set of pitches.
A tritone substitution is somewhat different, since it depends on what the analyzer thinks the chord could/should have been. There's a recent YouTube video (by David Bennett, IIRC) that shows tritone substitutions in songs, but he mentions that he doubts Paul McCartney knew he was writing one. But does that mean it isn't? No, it means you can see it as a tritone substitution.
Music theory is a subjective tool to describe and compare, nothing else.
I think this might be missing the point a little bit - the bassline in figured bass has more primacy than any of the sonorities above it. The most relevant feature of "chord"-based approaches to music (as found historically, e.g. in performance methods for the Baroque guitar, and to a far lesser extent in repertoire for the lute) is that the vertical sonority (such as "Cmaj7") is all that matters; individual "voices" or "parts" are of secondary importance at best. This maps well enough to how lute- or guitar-like instruments might be played in an accompaniment role (since these instruments obscure the sense of a "melodic line" or a "part" the most, for a variety of reasons) but music includes a whole lot more than that.
It's very clear that the man wrote chords. It's also very clear that harmony (ie vertical music theory) was not the tool used to write those chords. If you listen to BWV903, for example, there is a long section of arpeggiated chords that defy 18th and 19th century harmony.
The OP's idea of a tritone substitution from Scarlatti does not have the same harmonic function as a tritone substitution in modern music. I would agree with you that if the function matches, you can certainly analyze things in the context of a later theory, but the function does not match.
Finally, figured bass notates intervals, not chords. A crude realization can neglect voice leading in the upper voices, but more skilled realizers of figured bass will preserve voice leading between the intervals in the upper voices. The idea that figured bass is about chords sort of maps backwards the modern abuses of figured bass (which IMO should not be taught as a "basic music theory" subject - but that is for another day) onto the original method.