I'm still with Andrew on this one. I always thought that the transaxle casing was heavily ribbed for strength, not to cool the oil. Ever driven an older Jaguar, like an E type or XJS? The 'box is famous for cooking the driver's leg even in the middle of winter
The transmisson of any car gets hot in use, the issue here is duty cycle. Heavy loads (Westys on tour), excessive torque input and multiplication (V6/TDi syncros) and giving it the beans in G (muddy playtime) all push the transmission up it's duty cycle curve, necessitating more frequent oil changes before it degrades excessively and reduced service life between rebuilds.
Look under front of any moderately modern car and you will find that they all have undertrays for aerodynamics and noise suppression; also very enclosed gearboxes* usually with the battery tray and fluid reservoirs positioned right over the 'box to make use of the volume. *This is the case whether front or rear (four) wheel drive.
This is something that particularly interests me for our quicker and lower buses

might just pick up a few mpg and lose some dB in the cabin...
1985 Oettinger 3.2 Caravelle RHD syncro twin slider. SA Microbus bumpers, duplex winch system, ARC 7X15 period alloys