The main danger is if the community trusts such an organization too much. For example- if everyone assumed the client version put out by the organization was trustworthy, then there is serious danger. A group as you propose should probably exist, but the community should remain skeptical of it, and always constructively critical.
I thought of the exact same thing, now is the time to make one or more services which sole purpose is to constantly review the downloadable official client for changes, and alternative clients too. As long as these checks are in place, it should be difficult to sneak in backdoors. Such checks are no distrust, but rather a salute to the current developers, and the very nature of bitcoin.
And then if some person doing these checks turns out to spend an awful lot of time on it, he will end up having to make a Bitcoin Foundation Foundation, in order to raise money and have his salary paid by it, so he can do it fulltime. After which we will have to cry out at the fact that people who did something good initially (checking the dev's code), now have gone ego-crazy and started an undemocratic foundation that claims to be THE organisation that checks the dev's code! Then someone would have to start checking the work of the code checkers, until it starts taking him so much time that....