chris, your logic may sound great, but it has little relationship to the physics of audio recording.
Let's start by understanding some of the laws of physics (which cannot be changed.) Sound waves, just as light waves, decrease in strength by the square of the distance. So the strength (or relative loudness) of a sound picked up at one point will be only 1/4 the strength at just half the distance. I think everyone would agree that to get the best sound with a shotgun, the shotgun needs to be as close as possible (generally just outside the camera frame.) That is because the sound volume (or strength) falls away so quickly. The best recordings are made with the mic as close as possible to the source of the sound. Then simple physics will prove the loudest relative sound will be the sound you want to record. And that is where we switch to the physics of recording & playback.
There are two kinds of signal to noise ratios to consider when recording. The one most folks think about is the physical rating of the mic and recording media. Which is set in stone once you have selected your gear. There is also a signal to noise ratio for the sound in the room. Consider what happens you're listening to a friend while watching a movie on TV. Everything is fine until the commercial comes on and now you're having trouble understanding your friend. What happened is obvious to all, and would be described as change in the signal to noise ratio of the conversation. When the movie is quietly moving the plot along, the voice signal is high in comparison to the TV noise signal, we have a small signal to noise ratio. And small ratios are better, it is easier to distinguish between background noises & the speaker because the speaker is much louder than anything else in hearing distance. Now when the commercial blares out, the TV noise signal is suddenly several times louder. The resulting signal to noise ratio begins approaching 1:1 and at a ratio of 1:1, the noise & the speaker are at equal volumes making it very difficult to understand what is being said.
Suddenly we are back in our recording situation. A lav microphone on the chest of your speaker will necessarily have the tiniest signal to noise ratio physically possible. This has nothing to do with the fact that lavs are generally omni-directional and shotguns only record a narrow angle. When you are physically closer to the sound source you will have a smaller signal to noise ratio. And when recorded, what you will hear is only the sound you want to hear. With a shotgun mic, you will have a much weaker signal to begin with. And that signal has to be electronicly enhanced to even approach the signal strength of a mic on the sound source. And that adds noise. That is a fact of physics. Consider as evidence the mics worn by TV hosts. Even on network news programs, speakers are wearing lav mics. That isn't some accident, they know the best recording of a speaker comes from a mic on their chest. They also know that if you are talking to un-mic'ed guests, you need to place a stick or handheld mic near their chest. If it wasn't the best recording, wouldn't they use shotguns on set? Simply thinking about how pros operate should have made it obvious. The only techs who don't record with mics on the talent are working in conditions where a lav cannot be used for, usually, artistic reasons.
So now that we understand how the world actually works a little better, let's get to the question of room echo. I'm going to omit the description of what makes some rooms better or worse in terms of generating echoes. What we are concerned with is how does that echo get recorded? From understanding the physics of sound recording, we know that to be audible on tape the echo must be louder than other background sounds. Obviously an echo has less volume than the source sound, so when can the ratio of echo to background noise be small enough to hear? Well, it can be a very loud echo. But most of the time, the echo is occurring when our speaker isn't speaking. In order to do that, the echo has to be louder than room noise at the point the mic is operating. That is determined the physics of the room. But to be audible, the camcorder has to have audio recording levels set high enough for the fairly quiet echo to be louder than other sounds. As we learned, the farther we are from the source of the sound, the more amplification we require to get the same recording level on tape. Using the on-camera mic for a two shot at around eight feet away means the mic will require a higher recording level setting (or greater amplification) in order to "hear" the speakers. So at the mic location, the echo is also boosted in strength and easier to hear.
It is simple to demonstrate that the speaker to echo ratio increases at the square of the distance from the speaker. (In general, the room echo is at the same volume everywhere in the room. So as the volume of the speaker decreases due to distance, the relative volume of the echo is increasing.) There is no way to change this short of changing the echo characteristics of the room. So here's what's going on with the sound levels. The ratio of echo to noise remains the same everywhere in the room (for practical purposes) with the ratio of signal to background noise varying by the square of the distance. So at a certain distance from the signal source (our speaker) the ratio of signal to noise will approach the ratio of echo to noise. Effectively, the echo has become as loud as the source sound.
Now here's the tricky part. In order to record audible audio for playback, we adjust our audio recording level according to the strength (or volume) of the audio at the point where the mic is located. The less the volume, the higher we adjust our recording level to compensate. So the closer the mic is to the sound source, the lower we set our recording level. It obviously follows that we will record less noise (and echo) the lower we have the record level set. Without question, the loudest (or strongest) audio occurs closest to the sound source, so we can set our recording levels at the lowest possible setting. Which generally means the noise is too quiet to be recorded. Of course we have to be setting our audio recording levels manually for this to work best. Modern camcorders on auto-level respond so rapidly to volume changes that when our speaker stops speaking, the audio levels leap up to the maximum in microseconds. So you can still get noticeable echo or other noise whenever the speaker pauses. This effect is greater the farther you are from the source.
This is also true with shotgun mics. A shotgun cannot pick-up louder sound; it works by picking up only a narrow cone of audio and boosting the levels considerably. It can pick up less audio noise by proper aiming. You generally want to use the audio source to block the cone of audio pick-up. If you set your shotgun on the floor & point up towards your speaker's head, you can get really loud room noise mixed into the signal. Especially if you're running auto-levels.
So here's how it all shakes down in practice. Lavs mics have no competition in picking-up voices. Nothing can work better because of the physics of sound & audio recording. But to get the best results, you'll want to set audio levels manually. (Although I've found a lot of situations when I don't have an audio mixer available for two speakers with very different speaking volumes, auto-levels recorded superior sound. But part of that is because my VX2100 doesn't let me set levels independently of each other. On cameras with independent level controls, I always set levels manually.) For an example of a lav getting great sound in a noisy location, you can check out my iReport on CNN of the Des Moines Floods on Friday the 13th. (www.ireport.com/docs/DOC-35292) The second stand-up begins with a shot of water being pumped back into the river, then I start speaking. You'll notice I comment on the loud diesel pump in the background, but you can't really hear it. Where I was standing, the pump made it hard to hear myself. But the properly adjusted lav mic on my chest only recorded my voice. With a good shotgun operated properly, I might have gotten similar audio. But with an inexpensive wireless lav, I got great sound and could afford to buy a wide angle adapter for my VX2100.
So I hope this clears things up a bit. Simply stated, recording a speaker with anything except a lav mic is a compromise. With enough money & man-hours, you can compensate for it. This is why only folks who cannot use a lav mic are the only folks not using lavs. And why I'd recommend you try using lavs instead of your on-camera mic. I think you might be surprised at the improvement.
Good luck in your productions and have fun.