one of my biggest problems when combining melody with chords is that ending a melodic phrase by going M3 -> 1 is one of the strongest sounding melodic resolutions, but the strongest harmonic resolutions I know (V7->I, bII7->I, iv6->I) all do not work well with the M3rd of the tonic chord included (at least when the blending is clear enough to hear the dissonance). I know V13 is a thing, but I always think that sounds weird unless the 7th is lower than the 13th, which usually makes the chord sound muddy.  The best way I've found is iii with the 5th of the tonic scale in the bass, but that never is quite as satisfying as I want it to be.
if you are really saying the 3 to 1 melody is the strongest thing around, then surely you must have Heard it somewhere. how was it done there?