Odd, I train in the most competitive region for flyball and sub 4.0 dogs are common place. In fact the team I train with has broken the flyball team record over and again and currently maintain it. That said we have a TON of dogs that are cross trained in lure, dock dogs, obed, agility, herding, rally, and schutzhund.
It's an old school idea that training obed and control will squelch drive.
I like my dogs to have a firm grasp on one thing before I add in more, like training schutzhund and agility has confused my malinois in the A-frame. I ask for very different tasks with each frame so I have decided to hold off on agility until we get a 1.
However we cross train with schutzhund, flyball, and obedience. We do not cross train competition obedience beyond basic level because again, it can confuse the dog needlessly. That is not to say I don't plan to get every title I can but prioritizing what I expect my dog to be the best at is only rational.
Rally has shown for some to allow for sloppy handling which will cost you in schutzhund when it becomes habit. This is another I plan to start, time and energy allowing, but I will be asking people to keep a stern ear and eye to my handling. The last obed show I was at the judges all reminded everyone about this not being rally and talking to your dog will have you DQ'd beyond the alloted 2 command rule.
In flyball, when training a turn, we've found dogs trained for SV/EV have more trouble getting horizontal because they've been ingrained to reach up. This has totally shown to be an issue with Sloan and we're working on training a better bending of the body to compensate her reach.
I have found the issues with cross training during foundation work for my competition dogs are minute things however they all amount to greater issues if you're not keen to their happenings.
However, once a dog has a firm grasp on what is being asked I'm all for title whoring and variety. I wouldn't be as involved as I am in so many sports if I felt else-wise!