I'll give my two cents on this.
A shore bombardment ship is useful in a completely different sense then the class WW2 D-day style. What it would be used for now is medium/short range tactical strikes in a Libya-eske situation, in order to reduce the cost of Tomahawk spam. Therefore, nothing bigger then 8" would be needed, and 5" would suffice with extended range projectiles. This leads to an ideal ship, about 150-200m long, carrying a single 8" gun, a few 5" guns, and a 32 cell VLS for self defense and for Tomahawks.
You do have some things right here, such as the solution for Tomahawk spam. Mainly, I opined for major caliber weapons (larger than 8") due to the fact that VLS cannot currently be reloaded at sea. SeaTRAM was a project designed to reload them at sea, however, the USN shelved it for whatever reason.
It would be my opinion, than 8" guns would have the same issues with mitigating the Joint-Fires gap as do the 155mm and 127mm systems. You could remedy the issue by building perhaps 12-18 of these ships you conceptualized (remember, it's estimated that 21-24 DD(X)'s would be required to mitigate the joint-fires gap). And even then, you would miss out on some of the proposed missions that major caliber guns in excess of eight inches are proposed for dealing with (handling some missions previously carried out by aircraft and bunker busting bombs/missiles). Where as 2 ships with an Iowa-like armament could mitigate the Joint-Fires gap (you'd need 4 altogether, so that you could have 2 on station year round).
The cost savings with the larger ship armed with more guns and VLS would be quite obviously substantial. In this case, smaller isn't better. Then there's the issue with magazine capacity. A larger magazine capacity means more missions per ship are possible within a given time span, and it would be possible to be ready and on station, longer. However your ship would be around the size of a DDG, and thus would lose such an advantage by virtue of available space.
You'd need more of them. And if using a sabot, the projectile would be on par with 155mm or 127mm due to the diameter of the actual projectile being reduced (due to being saboted). And you'd lose some of the hard-skinned kills. After-all if you look at the OMFTS scenario listed in the Colonel's thesis, you see far more vehicle kills from the 16" guns than you do the 5" guns.
Also, you have further problems in the fact that you lose range with a reduction in caliber like that. For instance, An engineer with P&W stated a 16" ram-jet projectile was feasible, with a range roughly estimated at 400 nautical miles. DARPA produced a study for a mass-fraction projectile (also 16") with a range somewhere between 200 and 270 nautical miles. Upon seeing these numbers, it's prudent to reflect on the 100 nautical mile range of the 155mm/62 Advanced Gun System, and it's actual range so far achieved is only 59 nautical miles. Smaller caliber guns will have shorter ranges overall, even when using range-assisted & long range guided projectiles.
The fact is, is that while an 8" system does have merits, it would not produce enough of an increase in capability over existing 155mm and 127mm munitions. Therefore, it would be illogical for the USN to pursue such a project.