No actually they didn't. That's not a Sam Browne Belt in the first place, and secondly the Marine uniform shown in your photo is derived from an old Army uniform, from which "Mitchel" prototype is also derived. The "Arnold" prototype actually is cut more like your alpha, but is derived from a WW-II era Army Air Forces uniform. (Air Corps was the term used for the branch, like today's Army Aviation Branch, Infantry or Engineers. It had earlier also been used for the forces, but just before WW-II the HQ Army Air Forces was stood up, led of course by General Hap Arnold. Thus the Arnold uniform prototype.
BTW, this is a Sam Brown Belt
In this instance, worn by General Black Jack Pershing.
You'll also note that the uniform he's wearing is very similar to the AF "Mitchell" prototype shown.