Image Stabilizations only compensates for camera movements.

IS has no effect on subject movements. None. Zero. Nada.

If your subject is moving, only a shorter exposure time ("faster shutter speed") will reduce the amount of blur caused by the subject's movement.

Some related questions here at Photography SE for further reading:
What is more important, f-stop or IS
What reduces blur from camera movement more: large aperture or image stabilisation?
Is Canon 24-70 f/2.8L II that much better than Canon 24-70 f/4L IS?

Is there a stage at which there is a lag with the image stabilisation system that is overcome by shutter speed? There must be a time element within any IS system that impacts the image detrimentally as in rapid high speed shooting (7 or 10 fps) where the IS cannot keep up with the repeated shutter speed of 1/1000 sec?

IS in lenses such as the EF 600mm f/4L IS II keeps up just fine at 14 fps when used with the EOS 1D X Mark II. It's just that as the exposure time is reduced, the amount of blur caused by camera movement at the same rate of change is reduced by the shorter exposure time.

Keep in mind that regardless of the exposure time, as long as it is shorter than the time it takes each shutter curtain to transit from one side of the sensor to the other, the amount of time needed to expose one side of the sensor to the other does not change. What changes is how long each particular spot on the sensor is exposed as the second shutter curtain chases the first curtain across the sensor.

Notice that the shutter curtains cross the sensor at the same rate for the four exposure times shown: 1/1000, 1/2000, 1/4000, and 1/8000. The only difference between 1/1000 and 1/2000 in your 5D is that the very narrow slit between the first and second curtains is half as wide at 1/2000 as it is at 1/1000 even though the time it takes each curtain to cross the sensor is about 1/400 second in both cases.

When the EOS 1D X Mark II is shooting at 14 fps the interval between each frame is approximately 71 milliseconds. At sync speed or faster, each frame is exposed in about 2.5-3 milliseconds while any specific spot on the sensor is exposed for 1 millisecond at 1/1000 and for 0.5 milliseconds at 1/2000. That leaves about 68 milliseconds between each exposure when the mirror is moving down, the AF system and metering system are doing their thing, the mirror is moving up, the camera is waiting for confirmation that the mirror is all the way up, and the shutter is told to fire.

To understand just how little of the total time per frame is spent with the shutter actually open (partially at any given point with Tv shorter than flash sync speed), check out this video from which the above GIF is extracted. It's cued to a point right before the shutter button is fully pressed. Waiting for the mirror up confirmation seems to take forever. It's much longer than the actual exposure!

I have noticed that when shooting birds in flight, I get better results shooting at 1/2000 sec with the IS system turned off that shooting at 1/1000 sec with the IS system in use (IS used in either of the 2 settings).

The difference you are seeing between using a Tv of 1/1000 and using a Tv of 1/2000 is that the birds are only moving half as much in the 1/2000 second their image is projected onto a particular spot on the sensor as they are in 1/1000 second. It has nothing to do with IS.

In terms of lens alignment, IS always has the potential to be detrimental - even at slower shutter times. With lens based IS, the movements of the IS element/group introduce mild misalignment of the lens. This is viewed as acceptable if the blur introduced by the less than 'perfect' optical alignment of the lens¹ induced by an IS movement is less than the blur that would otherwise be introduced by the motion of the lens/camera. Eventually, as exposure time is shortened, one does reach a point where the penalty of slight lens misalignment outweighs the benefit of compensating for blur caused by camera motion. That does not, however, explain your observations with regard to birds in flight.

As with almost everything photography related, there isn't any "free lunch." For how both IBIS and lens based IS can affect image quality, please see this answer to Does IBIS reduce image resolution? How does it compare to lens based IS?

¹ There's no such thing as a 'perfectly' aligned compound lens, even among non-IS prime lenses. There are always manufacturing tolerances to be considered.