I have a few issues with the mod.
1) I like the Ithaca Model 37, particularly the Featherlight, and what he did is something I regard as abusing a firearm.
2) The problem with the entire theory is that you are adding a fair bit of weight to the gun and doing it in a manner that not only ruins the lines but also the handling.
3) One of the major factors in choosing a shotgun is how well it fits you. The drop at the comb and heel in particular are vital in terms of getting a shotgun that actually points where you are looking when it's mounted to your cheek. He's taken that option right off the table.
If he really needed to carry rounds on the stock, a slip over carrier would have been just as effective, and would not have seriously impaired the ability to actually shoot the shotgun accurately. However, my preference has always been to carry shotgun shells in a pocket or a belt. When I was a kid I bought a never issued M1 Garand cartridge belt. Each of the 10 pockets held three 12ga or 20ga shells.