This cutaway diagram is of the 5D shows clearly the small autofocus mirror behind the semi-opaque main mirror. Towards the bottom of the camera is a spectacle-type lens which focuses parts of the main image onto the AF sensor.
( credit to canon.com for the original image from which this was cropped )So you can imagine how a flapping main mirror causes the AF system all sorts of issues; it can't reliably detect out-of-phase conditions whilst the mirror is in motion and this has a direct effect on the burst rate. For this reason many cameras have options to allow shutter to override AF in burst mode, or vice versa.
Looking forward to a good mirrorless camera in the near future!