We do something similar, but after 20yrs at this house & a million animals, I'm starting to run out of room. Plus I'm getting tired in my old age & the clay/soil in my area is hard digging. I buried my latest bird yesterday (RIP Hey Hey.) Fighting the soil in sweltering heat I just kept thinking, "theres got to be a better way." It gets too hot here to have anything in the trashcan, but after an hour I seriously started to consider it. (I hope my DD isn't reading this.)
I can't do much else. Like you, it gets far too hot for digging in the day so I often do it at night. The sheep call me the grave digger!
I can't put them in the compost because everything is pretty much free range here and of course the compost heap is on everyone visit list.
I suppose there will come a time when I do what I do with eggs that have been sat on but not hatched which I take further up the mountain, brak open and leave for whatever finds them.