1) You don;t need AI for a "gray goo" scenario. All it takes is a self-replicating nano-disassembler that is able to reconstruct extremely varied source materials into energy and copies of itself. Intelligence doesn't factor into it, any more than a natural virus or bacteria needs to be smart to eat everything it's able to eat (yes, I know viruses don't actually consume, the point still remains
This is fortunate, because a nanomachine like that is going to be difficult
enough to make without having to add some method of intelligent control. Fitting a method of communication for each nanobot to send and receive information to participate in and receive instructions from a communal intelligence into something that small while still allowing it to perform its self-replication function boggles the mind.
2) Nanotech is
slow. When you're only a few nanometers at
most, you aren't going to be moving at high speed. In fact, it's difficult to conceive of a form of propulsion that could be used by a nanobot other than the examples we've been given by bacteria and the like. If the nanobots are actually in a "gray goo" scenario, then they aren't even going to be spending most of their time and energy moving - they'll be consuming whatever is close at hand and making copies of themselves. And ripping apart molecules and reassembling them an atom at a time as proposed in most nanotech scare fiction
is not fast.
Yes, they'd be replicating at an exponential rate. But they're not exactly going to cover the Earth within days or even weeks or months, any faster than bacteria grows in a petri dish. This leaves them vulnerable to any form of countermeasure we can come up with.
Worrying about nanobots being poorly designed and attacking unintended targets for self-replication (attacking healthy cells along with the cancer cells, for example) is a very legitimate concern, and it's something that engineers will be stressing over when we reach the point where such technology is really feasible.
Worrying about a "gray goo" covering the Earth and replicating until there's nothing left except nanobots is hysteria. Like most scenarios that involve the End of Life As We Know It (tm).