Yup, that's pretty much it. The trick doesn't quite work with chords in that way.
When you're harmonizing with thirds, you have to chose between using major third or a minor third. You (usually) pick whichever one gives you a note that's in the key you're playing in.
If you wanted to extend this concept to chords, you'd have to account for all the notes in the chord--not just the root note. You'd need to use the appropriate chord that would contain notes in the key you're playing in.
So if you're in the key of G and someone plays a C chord, you could play some type of E chord over that, but not an E major. You'd probably want to play an E minor.
Basically, you'd be harmonizing using the appropriate third above each note in the C chord. Think of it as doing the "thirds trick" three times at once!
C major is C, E, and G. What would you play over each of these? Well, you'd play an E note above the C, a G note above the E, and a B note above the G. The result would be E, G, and B, which is an E minor chord!
(And when you take the notes of the C major and E minor chord together, you get C, E, G, and B, which is a C major 7th chord!)
JJ