Agreed!
I never use Melodyne or other plugins for harmonies, I always sing them.
More often than not I double the harmonies and pan them left/right to give them depth.
Harmonies are certainly not always thirds, it depends on the melody line, the chords and the effect you want to achieve.
Some of my harmonies follow the melody and some are ‘colour notes’ (ninths or sixths or other) that may or may not follow the melody. A good example is the song ‘Yesterday Fell’, on my website: http://www.hornforhire.com/music.html
Experiment a lot, and use what you like!
It may take some time, but eventually you’ll get better at arranging (and singing) harmonies.
Good luck!
Wim