Why would anyone use a shotgun in field combat operations in modern warfare?

My understanding is that the M14 was popular with second shooters/observers in a sniper team.

Well, I've never heard of entire platoons in Vietnam armed with shotguns, so the M16 must have been good for something.

Anyone who thinks a 12 gauge pump shotgun can hold its own in a firefight against AK47s, SKSs and RPDs, well I don't know what to tell you.

"Be advised in the REAL WORLD the M-16 had a nasty habit of jamming, usually in the middle of a firefight."

Only the early ones, but that problem was solved by the time the M16A1 was adopted in 1967. The M16A1 was/is as reliable as current M16/AR-15 family rifles, and they are very reliable. They do especially well in mud tests; better than any other well-known military rifle, including the AK-47 family.

There are many known cases of AR-15s going thousands of rounds without cleaning, and without any malfunctions.

A properly manufactured AR-15 which is fed appropriate ammunition from magazines which are not damaged or otherwise out-of-spec, is at least as reliable as anything else out there. The problems in Vietnam which people still wrongly blame on the AR-15 itself, were due to out-of-spec ammunition, and exacerbated by the lack of chrome-lined chambers, which itself was exacerbated by the lack of cleaning kits.

The lack of cleaning kits wasn't a problem because the AR-15 needs to be kept meticulously clean in order to be reliable (it doesn't need to be), but because, over time, and especially in the Vietnam jungle environment, it accelerates rust, because the fouling particles retain moisture. Chrome-lined chambers would have greatly reduced or eliminated that problem, though there still would have been the out-of-spec ammo problem (it was out-of-spec due to inappropriate powder with the wrong pressure curve, which greatly increased the cyclic rate, which resulted in the action trying to extract the cases while they were still pressurized against the chamber wall, resulting in failures to extract). Chambers which were pitted from rust made matters worse, because the brass would expand into those pits, making it even harder to extract.

Take a brand new Colt model 601 (the first AR-15 that was used in Vietnam), feed it ammo loaded with the IMR stick powder that it was designed for, and it will be perfectly reliable, even in the jungles of Vietnam, even without cleaning, for the first several days at least, no matter how many rounds you fire. However, if you continue to neglect cleaning and oiling it, after a few weeks the non-chrome chamber will be rusted/pitted, and then you'll probably start having some extraction problems. Switch the good ammo for some ammo loaded with inappropriate, cost-saving WWII surplus Olin ball powder reclaimed from artillery shells (which is what actually happened), and then you'll definitely have some problems.