There is definitely problem with this animation missing. It makes game looking really dated I think. The AI often when taking part in firefight to move from cover to cover will just stand up from prone, make few steps and just then start running. Specifically it would be great for AI and also would improve the immersion.
There should be some type of prone-to-run animation that would be used in combat behaviour for AI - as well as for the player, so we could automaticly or by some button combination (forward and up-stance button? 2x forward button ala A2? not sure about that) like push off the ground and start running without taking risk of being shot and make it into the cover.
That should give few seconds advantage but also little bit more fatigue too.