Flex on using a tripod, increased chance of vibration (FoCal recommends removing the grip during AFMA), and risk of the tripod mount popping out when using a Blackrapid strap are three main reasons why I never considered a battery grip.
Also, using L brackets with grips reduces the flexibility of taking them on and off. You can also add weight as a factor, especially using f/2.8 zooms and a full-size Speedlite on top.
Recently, I bought a 5DII with grip (just because it came with it) and tried the combo for a few days. I realized it was more awkward for me to use the grip in the portrait orientation- the grip causes my hand to be farther from the center of gravity, and higher than the plane of the lens. I found it was more comfortable to use the regular grip over the lens. The grip found a different home, and I suppose I shall never be contemplating a gripped body.
[Sidebar: this discomfort was far less with a 1D or 1Ds body I tried briefly last week]
They could be made more rigid if they only supported 2 batteries rather than the (rather useless) AA battery cartridge. Without that a web could be fitted between the two batteries to reduce the unsupported length of the base! Anyone ever use the AA battery cartridge?
Both the battery grips for 6D and 5D III have longitudinal battery trays for the LP-E6s, instead of vacant space. So the BG-E13 should have been less wobbly in theory.