Audio expert Bob Heil, who pioneered high quality concert sound for the Grateful Dead, The Who, and more, always preached the virtues of microphone phasing. No amount of EQ can make up for sound that isn’t there.
As an experienced production guy, I say you're absolutely right. If the sound being captured is sh*t, no matter what's done to it via EQ, compression, verb/delay, it'll never be right.
In the studio, for instance, a clean, great sounding raw track will sound great with a minimum of effort during mixdown.
In the studio, I've pulled my hair out trying to massage badly recorded vocals, guitars, basses etc. into shape. Most of the time the solution is to record them again - correctly.