Just some general info about what I do for lubing them type lawn tractor transaxles.
On them MTD single speed FNR lawn tractor gear boxes when I opened them up for repairs (for myself) I just cleaned out the old caked/gunked up clay grease and repacked with EP2 wheel bearing grease, especially the axle bushing housing and the ring gear area. One reason that heavy grease is used, the axles tubes do not use any oil seals. It's common for the outer axle bushing to wear out but the inner bushing still be ok. If the tires appear to be tilted out at the bottom when viewed from behind the mower the outer axle bushing are worn out due to lack of lube. If not worn really bad the inner and outer bushing can be swapped so as to give good axle support at the outer end. Replacement bushing can be found if desired. Usually the inner axles busing are ok, but the axle support needs to be held to a minimum tolerance to reduce change of spider gear breakage I also have added little bit of 90-140W oil to the center housing area. Not enough to flow out the axles. I've also drilled a hole in the center aluminum housing and add little bit of 90-140 weight oil and then seal the drilled hold with a screw. (add just enough oil that it does do get about the lower level of the axle tube)
I've had really good results doing this lube methods.
I also suspect that the 2nd reason that MTD used the red clay grease that becomes hard packed is it allows the lawn tractor's wimp built single speed transaxle to wear out faster.Summary: Most any type of grease lube is several times better than the OEM clay grease for this type gearbox. The 90-140 that I add is mainly for the center ring gear, spider gear area and shifter fork area lube.
Not a fun job taking 2 or 3 of those wimpy made gear boxes and making one.