I'm sure some have - and do - compete with MIL-Spec guns and I've known a few who started out that way, but darned few 1911's stay MIL-SPEC for long once the shooter starts shooting a lot!
A pure MIL-SPEC 1911 is not very ergonomic and most folks find them pretty uncomfortable to shoot under IDPA match conditions (which closely simulate concealed carry conditions). They were designed for warfare not concealed carry or action pistol competition. Their role as a weapon of last resort, in the hands of a soldier, drawn from a flap holster and fired sparingly, is an unquestioned success. But the MIL-SPEC 1911 was never intended to be drawn from concealment with speed, safe-off on the fly, rapidly acquire sights, execute rapid double taps and speed reload!
The major modifications made to IDPA competition and concealed carry 1911's consist of items intended to improve the overall handling characteristics of the gun. Better hand position via the upswept beavertail grip safety allows faster follow up shots with more comfort and promotes proper hand position even during a rapid draw. The extended thumb safety allows fast, positive, one handed engagement and disengagement. The beveled mag well (or even add on mag well) helps with reloads during the stress of competition (or a gunfight?). The flat, checkered mainspring housing improves grip and pointability.
Then there's fit - the tighter fits on todays 1911's DO NOT affect reliability if properly done - there's just too many that aren't properly done! But that is a manufacturing quality problem, not a design flaw! There is no need for a non-military sidearm to be sloppy. The mission is different and the conditions are different. The sheer number of rounds fired through a typical IDPA gun (and hopefully a carry gun) demand tighter fits and proper care in order to last. The improvement in accuracy is required for the new mission of the old warhorse - NO COLATERAL DAMAGE!
And the fancy shmancy sights are a heck of an improvement over the little bumps that grace the slide of a MIL-SPEC gun!
If you still have your heart set on a MIL-SPEC gun, you can get real close with the Springfield MIL-SPEC model or the Colt 1991-A1. Both are their manufacturer's attempt to go "retro". You'll have to cruise the gun shows or find a private supply for a real G.I. piece and finding one "un-tweaked" may be difficult.
Mikey