A solid core acts as one big fat single turn shorted secondary winding. Do I need to say anything else?
OK I will.
Using laminations that are lacquered to prevent conduction through interconnecting layers of lamina is the only option. The small eddy currents induced in each laminate are basically a tiny, tiny fraction of the vast current taken by a fully conducting core.
There is no option but, if I theorized on a great magnetic material that was non-conducting, then it would likely have less permeability than iron. Hey they do one - it's called ferrite but it has less permeability than iron and saturates at a lower flux density.
Regards your thought: -
I would have thought that, at least as far as flux density goes, a
full cylinder-type core with a reasonable diameter would be best.
Magnetism is caused by current and if you have multiple turns of wire the magnetism is multiplied by the number of turns and this is called magneto-motive-force (MMF as opposed to EMF) measured in ampere turns. Magnetic field strength (H) is the daddy regards transformers and is MMF/metre where the "metre" part is the effective length of the core.
So far we have H and no mention of core permeability. To get B (flux density) you use the formula
B = \$\mu H\$
\$\mu\$ is the effective permeability of the core so, the higher the permeability (for the same current) the more flux density you get. More flux density means a higher likelihood of saturation so actually, as you make a transformer core more "magnetically permeable", you are potentially making it saturate more.
This is why a lot of transformer applications place a small gap in the core - the onset of saturation is lowered but more turns are required to achieve the same inductance and then copper losses start to be significant. It's a balancing act.
Hope this helps.