I have shot plenty of 30/06 bullets. I have never noticed it to be inherently inaccurate. Same for the .308 win of course.
“... the ability to shoot the heavier bullets gives it an advantage; the extra case capacity gives it an advantage; the ability to stick a chamber insert into a 30/06 chamber and shoot the NATO round out of a 30/06 rifle gives it a big advantage in terms of ammo availability; the designation as a hunting round also gives it an advantage as far as ammo availability goes. ...”
Let us first be clear on military versus sporting applications.
Few civilian users care about the greater length of the 30-06 cartridge versus the 308, but the military cares a very great deal about such details. Pure considerations of rifle action length rarely bother non-military users, but it is an accepted engineering fact that any given receiver design can be manufactured shorter for a shorter cartridge, hence stiffer for a given weight. Stiffness improves accuracy potential. There’s a reason that fewer serious competitors use 30-06 rifles every year, compared to 308 (or 5.56mm).
The 7.62 NATO/30-06 chamber insert was developed for the US Navy, to convert M1s to the shorter cartridge before M14s were available in any quantity for shipboard arms lockers. It was abandoned in very short order, as it was found that no one could make the insert stay in the rifle, except by radically altering the original chamber (FN provided the “final solution” by reaming out the 30-06 chambers and installing oversize inserts, under cryogenic prep if I recall correctly). Such conversions rendered the rifle permanently incapable of firing the original cartridges. Most experienced people recommend avoiding the inserts today, for safety reasons: they have a bad habit of staying with the shorter case on extraction, even in bolt action rifles. No practical advantage in terms of ammunition flexibility exists, set against that sort of safety risk.
Every military organization abandoned the heavier bullets (180 gr and higher in 30 cal) before 1940, for standard issue rifle cartridges (US, Germany, France and UK did so well before WWI). They simply could not be driven fast enough to produce a usefully flat trajectory, and their effective range was actually less than the lighter, pointy bullets. “Full power” rifle cartridges were abandoned as military rounds after 1930 because average troops simply could not estimate range closely enough to exploit the extended ranges possible.
Note to shooters: even little bullets like the 5.56mm run out of effective range because the trajectory becomes so curved that few users can score hits. Even tiny errors in range estimation cause the projectile to fall short or go over. The bullets still have plenty of steam and can do serious damage at extreme ranges (4000m plus), but that matters little to military users, if there are no hits.
Nearly every military organization is now equipped with heavier weapons, that offer higher hit probability at range, fire warheads that can do much greater terminal damage, and can be controlled in action far better than can individual troops armed with rifles. No matter how effective an individual might be, armed with an older style rifle and “full power” ammunition, it is simply no longer worth it in a military sense.
None of this applies to an individual armed with a rifle, for hunting or defense situations.
Heavier 30 cal bullets find US military application nowadays only in specialist situations, as with the Army’s Marksmanship Training Unit and its program for snipers armed with 300 Winchester Magnum rifles.
Incidentally, AR10 platforms cannot chamber the long standard-head cartridges like 25-06, 270, or 30-06; they are simply too long to fit. 7.62mm NATO barely makes it, and the platform still has to be enlarged above the M16 size receiver components. 300 Winchester Mag is out of the question.
Very short centerfires like 22-250, 250 Savage, or 300 Savage still won’t work in an M16 dimensioned rifle: case head diameter is still “Mauser standard” (rim size 0.473 inch), hence too big for an M16 size bolt. Upper limit for an M16 size bolt is around 0.422 inch, that of the 6.8x43 or 30 Rem.