Here are some other things to think about:
The best way to secure the vehicle by the frame has the straightest run. If you look at the picture of the Olds above, you will see that the rear straps are almost parallel to the vehicle and trailer. So, as the car moves up and down the strap isn't getting much tighter and looser. If you secure straight down to the frame to you place HUGE shock loads on the straps/chains as the vehicle bounces. The only way around that is to secure it so tightly that the suspension is bottomed out.
Securing the axle is superior is this fashion, since the axles tend to stay right where they are, so nothing tightens or loosens going over bumps. However, as already mentioned, the vehicle is still free to "float" around, which can make the trailer less steady. A heavy vehicle with weak shocks bouncing around on the bed can be an issue.
Something that has become much more common in the towing industry are wheel straps. This is mostly because bumpers and air dams are getting lower and things are tighter, so it is much more difficult to secure cars to flatbeds without damaging something than it used to be. You can't even secure to the axles on most new vehicles, because so many parts have become lightweight, otherwise known as "just strong enough not to break when you drive".
Wheel straps grab the tires from behind (the inner sidewall) just above centerline, with another "dogbone" strap to create a basket, and then hood to the bed forward and aft of the tire. This is done to all four wheels.
Since most vehicles now have the tires close to the side of the body this means the straps will not touch the body of the car, and the car can not move side to side. It also should not move fore and aft, but it's always a good idea to lock the wheels somehow.
Unfortunately, I don't know of any trailers that are set up to use that tiedown method.