BosnMate beat me to saying all the parts you found are from a McClellan addle... except MAYBE the toothed buckles. I don't recognize them as being from the civil war Model-1859 McClellan saddle. There were more models of McClellan saddle issued by the army after the end of the civil war, into the early-1900s. For example, there is a Model-1904 McClellan saddle. A lot of its "hardware" is brass, which was mostly iron on the Model-1859. I hate to say it, but I think the parts you're finding are from a postwar model, probably the Model-1904. A lot of those were sold to the public when the army mostly got rid of its horses in the 1930s.
Note the heavy brass squares and short rounded brass eyes on the Model-1904 McClellan saddle in the photos below. Compare them with the iron hardware on the Model-1859, seen in the "floating" (all-white background) sideview photos. If you're not sure which photos show the Model-1904 and 1859, the model-years are shown in the photos' filename. To see it, hover your cursor over the photo.
A key ID-clue: There are rings at the front of the saddle, connected to it by 2-screwhole eyes. On the Model-1859, the connector between the screwholes is an iron sleeve. On the Model-1904, it is BRASS in the form of a loop like a button-loop... which is what you found. The photos BosnMate posted appear to show a Model-1904. You can clearly see the "loop" form of the brass 2-screwhole eyes in some of those photos.