I think you are over-complicating something quite simple. The 3-chord trick simply means that the chord of the first (or tonic), fourth (subdominant) and fifth (dominant) of each scale contain between them all the notes of that scale, so any tune played in that key can be effectively accompanied by just these three chords and no other [you can intro others if you wish, but it is not strictly necessary].

In the key of C, therefore, any tune can be accompd by the 3 chords af C [tonic] F [subdominant] G [dominant]. An added flattened 7th is often added to the 3 notes (1st, 3rd, 5th to form the major chord) of the dominant so the chord would be G7 - but this is not essential, tho it does give that 'anticipating return to the tonic' effect.

In G, it will be G C D; in F, F B♭ C; in D, D G A;   & so on —

Once you know this, it is easy to work out the first, fourth and fifth of any scale to tell you which chords will produce this '3-chord trick' accompt to a song in that key.