Makes me wonder how many of our old tractors have gunk like that in them.
IMHO petro based solvents are detrimental to ones health to some degree. For most of my cleaning I have gone to hot soap and water. It can take a bit more elbow grease but it works. But for a job like this I would not want to try it.
Another alternative for cleaning is gasoline. It cuts grease better than diesel but then there is the problem with fumes and fire. I have been thinking of adding gas to diesel for cleaning stuff like this. Yup not EPA or safety nazi approved.
It's a bit late but using gas to clean parts is way too dangerous. It's a good way to leave this earth or spend your remains days disfigured or worse.
For cleaning that transmission, I would use a putty knife and a roll of cheapo paper towels. Scoop, scrape and wipe as much as possible out, then use diesel or kerosene(whichever is cheapest) with a parts cleaning brush, clean it well, then more paper towels and wipe it down with oil until I got it back to work on it again.
I've been a volunteer fireman, using gas for cleaning and starting brush fires is bad news.