Rich, SAE standards for 10W-40 oil call for the same film strength at engine
operating temps as 40W oil.

To be sure, some will argue whether the standards are accurate or not (there is
room to argue).

That said, I too would use 40W in a transmission over 10W-40 unless I ran out
of 40W. A transmission runs cooler than an engine and is oil bathed so has no
issues with lack of lubrication upon startup.

Nope - common misconception!
When you're engineering a gear set, you need to select the correct film
strength so that the clearances BETWEEN the epicyclic surfaces do not
come in contact. A gear set typically is an unpressurized application.
The more power being transmitted the higher the film strengh needed -
and there is a correlation of (kinematic) viscosity (viscous shear) and
what it will support in load without "squeezing out" and permitting
metal to metal contact. ...... totally unlike a pressurized
application like a hydrodynamic bearing journal, although the
mathematics is the same.