Get the mic off the camera was a well-worn cry in the days of film and more recently tape transport which was picked up and there's nowt you can do about it later.
With off camera a good sound-man can get in close (whilst remaining out of shot). Certainly a radio-mic has no sync issue but only prosumer cameras have mic-input and I saw some reviews of the Sony VG10/20 that showed their audio-handling wasn't exactly friendly.
If you can't use camera mic-input....then maybe a hand-held Zoom-recorder - the more expensive versions having XLR i/p so you can use pro gear - the SD card just goes into yr PC - job done.
I think the almost ritual comment of get the mic off camera is still valid, but nowadays for different reasons. Small, but good mics were always a problem - so quite a few camera manufacturers simply added smallish cardioids to their larger cameras and they did work. Downsides were that fingernail noise was a problem when trying to find small buttons and switches near the mic - gain cranked up a bit, usually because of agc, and a teeny-weeny hunt for the white balance or shutter button would make very odd noises. Same thing with full size, separate lens style cameras - your right hand would be right under the better quality mic. Now the cameras have shrunk even more, so have the mics. I have to say that on the tiny pocket size Panasonic SD-9 I use for convenience shots, the 5.1 sound is actually very good. It's clearly not 5.1 as we know it as in post - the left/right v centre channels are not hugely different and as for the rear channels? What it does have though is sound that is quite useful once you list to the individual tracks - I often use this camera hung from the lighting bar on a theatre balcony - as a static camera. The image is great, and the sound quality surprisingly good - and the mix of the internal mics provides really effective audience cover, and also a nice stereo image of the band if they are live and not amplified. Obviously, and I think this is the killer - if you use this to record somebody talking to the camera from as close as two metres, it always gives a nice reproduction of the space, with the speaker somewhere in it. This is why separate mics are essential for this style of shooting. This is where I reckon the modern 'get it off the camera' comments comes from. It's NOT that the modern on-camera mics are bad - quite the reverse, they are too good at recording what a listener at the camera position can really hear - and this when reviewed is wrongly called bad sound, when it's usually high quality 'wrong' sound. When we had linear tape and then even h-fi camera sound, we often had noisy pre-amps and sub-standard mic elements that didn't work much at the bass end - so the slightly HF focus made voices a little better than music. Now we get better mics, but their result is deemed poor, which is a bit unfair. A ten quid omni lav clipped close to the presenters mouth sounds hugely better than a £500 mic too far away.