On September 19th, 1918 the Germans sent along the following message to the US Secretary of State:

"The German Government protests against the use of shotguns by the American Army and calls attention to the fact that according to the laws of war, every prisoner found to have in his possession such guns or ammunition belonging thereto forfeits his life. This protest is based upon article 23(e) of the Hague convention respecting the laws and customs of war on land. Reply by cable is required before October 1, 1918"

The quote from article 23(e) is a prohibition of "...arms, projectiles, or material calculated to cause unnecessary suffering..."

US Soldiers appeared to be aware of this note shortly after as there are accounts from officers saying they warned their men not to be caught with the gun in hand when surrendering.

Acting Judge Advocate General Samuel T. Ansell reviewed the matter and issued a lengthy opinion. The crux of his findings were centered on a simple comparison between the shotgun's blast and a shrapnel shell's burst. Both meant not to maim, but to kill multiple targets at once.

So the US would send back their own message. Basically "Do it and we'll begin reprisals."

The firm response must have worked because there are no documented executions of US soldiers for being armed with shotguns.

Now, the "why" of German disapproval is difficult. Their propaganda machine was set to work against the incoming Americans and would accuse them of being savage, tomahawk-wielding brutes with no respect for civilized warfare.

But you're right, it's hard to find the shotgun any worse than a machine gun. I think they were just irked to find another new problem to contend with and likely sought a diplomatic path to remove it from the battlefield.

As a follow up question:

What are some good resources to read about the Hague Conventions which banned expanding bullets?

I'm interested really in the difference of thinking between the goal of shooting someone. The ultimate goal is to remove that soldier from the field (I assume)? Is it that a hollow point to the arm would be more damage to the soldier than was necessary?

Because a .308 to the chest is probably going to kill, a .308 with a form of expanding tip to the chest is going to really make someone dead

Were flamethrowers widely used specifically as a weapon though? I thought they were mostly used to burn down fortifications and the like, especially considering how dangerous they are to use.