The thing about harmonies is... It isn't often that close-voiced four part harmony works very well, even with REAL singers. The trick, just like horn voicings, string voicings, you name it, is to spread the voicing wider. But this will often stretch a pitch transposer further than it wants to comfortably go.
Especially if you are a baritone or lower. You have to make sure all the harmonies are higher than your voice. Most pop music, at most you'll hear two harmony parts and the lead (if they are blocking around the melody). Often just the one. Try for a simpler harmonizer setting, and you may up with less mechanical sounding backing. One, maybe two voices ABOVE your singing, cassp, might help.
But, of course, good pitch is important, but flies away from the need to phrase like Van, who is anything but precise in his pitch! What's a boy to do?
For me, sadly, until these things can totally fool ME (I don't care if anyone else notices!), I can't bring myself to use the harmonizers...
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An arranger is just a tool. What matters is what you build with it..!