i always go to the axles because:
1. the chain angle is flatter to the trailer due to the lower mounting point.
2. if chained to the frame, the suspension can compress taking the weight temporarily off of the tires, allowing them to slip side to side.
3. chaining to the frame also lets the vehicle move front to back when the suspension is compressed and the lenght needed becomes shorter.
4. the above mentioned issue with the straps being able to actually fall off.
5. tow trucks always go to the axles with hooks. and they do it all day long.
you should always have some measure of keeping the binders from coming open. either wrapping the extra chain around the binder a few times, wrapping bailing wire around it, zip-tie's, ratchet straps or even using duct tape it thats all you have.
i also always put an extra chain or strap to the frame as well, with no pressure on it, because it is required by law for tow trucks to do this, so i figure it cant hurt.