OK- It's live. There are no rules. There are only tendencies and guidelines. I was trying to lay down a guide track, basic acoustic and vocal, and I needed it in a hurry for overdub people. And, damnit!, I just couldn't play the thing without singing! Yeah, I know- mental block. Need better chops. Anyway, I *plugged* the acoustic into a preamp (not generally recommended for recording- use mics) direct, and got the guitar track, then overdubbed the voice. I figured I'd mic it up pretty later after I worked on those chops, but damn! The mixing engineer listened and said, Why re-record that? It's fine! For anyone who cares, I used a Taylor 714CE with a Fishman Matrix stereo blender into a Joemeek twinQ, pre-EQ'd like a mother.
OK- one mic. If you are going to do that, mic placement is everything, and a good room is really important. Every player and singer has a balance, and separate mics/pickups are used for the nice guy with the board to smooth out those balance issues, as well as control tone with EQ. Other effects may be applied, usually reverb or delay, and occasionally compression. Later, in the mixdown stage, additional compression or EQ, noise reduction, etc., may or may not be added, according to the mixing engineer's preference. Is it still "live" if you cleaned up some noise? In my mind, yes. But- do we want to add "Cathedral" reverb, and make the guy sound like an Irish folk singer? Maybe not. I've run into similar issues recording choirs, when the director wants that fakey reverb sound.
Find the place, and the direction the mic is pointing, where the balance between the instrument and the voice sounds best to you when played back on speakers. That's called the "sweet spot". It's the holy grail, and I'll warrant many of us here have been seeking it for many years.
The short form- One mic is the wrong way to do it. I'd use 2 mics, accept that there will be "bleed", embrace the bleed, and just do it. Not all bleed is bad. It can often help to present a more realistic stereo model than totally discrete tracks. After all, what do you hear in one ear that you don't hear at all in the other? Answer-Something in headphones *that isn't real*. If the balance is already pretty good between voice and instrument, you can use spaced stereo, ORTF, coincident (X-Y) or other methods. Whether you put the two mics close on vocals and guitar, or use more distant forms of stereo recording, IMHO, you need 2 mics. You're not getting the answer you want because one mic is suboptimal. You have 2 ears. Start by training them to know a good room from a bad one, and find a good room to do it in.
P.S.- If you don't have the hardware/software, etc. to get 2 channels, find a way to get it. You need a 2 channel interface, 2 mics, and some software. Just do it. Resistance is futile. Good Luck.