Why does the liver produce ketone bodies instead of exporting acetyl-CoA from beta-oxidation for use elsewhere?
This is simplified version of a complex situation, but in summary:


A key role of liver is to control the distribution of metabolic fuel
for the other tissues (it always has enough for its own
requirements).
The way it behaves differs in the fed and fasted states (because of control by hormones and the concentration of metabolites).
In the fed state (after storing some sugars as glycogen) the liver will synthesize triglycerides and export them as lipoproteins to the adipose tissue, which will take them up and store them. (Other tissues in the fed state will be able to obtain glucose or fatty acids from the blood.)
In fasting and starvation the triglycerides in the adipose tissue will be broken down to fatty acids and glycerol, much of which will be used by tissues such as muscle.
The priority of the liver in starvation is to provide fuel for the
brain and nervous tissues in the form of glucose, by gluconeogenesis. The substrates for net gluconeogenesis under these conditions include glycerol and some amino acids derived from breakdown of protein. 
Other amino acids, produced in protein breakdown lead to acetyl CoA,
which cannot serve as a precursor for gluconeogenesis. However it can
be converted to ketone bodies (acetoacetate and β-hydroxybutyrate)
that the brain can use to replace some of its glucose requirement.


With this in mind, it can be seen that the reason for producing ketone bodies is not to generate a molecule that can cross the cell membrane into the blood, but to generate a molecule with special properties, lacking in fatty acids or acetyl CoA, that allows its use by the brain in life-threatening circumstances.

Of less importance, but worth mentioning, is the fact that citrate is used as an export form of acetyl CoA (when there is sufficient oxaloacetate available), a surrogate role it plays in any case in the transfer of acetyl CoA across the mitochondrial membrane for fatty acid synthesis. So it is not necessary to export acetyl CoA, and perhaps preferable to it keep intracellular.

A more detailed account of this is available at NCBI bookshelf online in Berg et al. sections 30.2 and 30.3