Well done Ken, you've done the research so you can proceed with confidence.
The reason I did not expect the same lubricant to be pumped through the hydraulics and the axle, is that the hydraulics need ultra-clean oil, and the axle dirties it up. The filter is an imperfect solution, because it can become choked up with crud from the axle. If that happens, either you have a pressure bypass (as cars always do) so fluid will continue to flow but will no longer be filtered, or you don't have a bypass, and no more fluid flows through the hydro unit. (Car engines would not do well if oil ceased to flow through them).
I had a Mini too, Ken. Bear in mind that it only had a car engine to protect, not a hydraulic pump and actuator, which are far more sensitive to dirt. The major source of dirt is blow-by in the engine - it is even dirtier than the manual transmission. Also, the Mini had notoriously low mechanical reliability - it was the least reliable car I or my friends ever had, and many of the failures the car was noted for were major mechanical. In particular the idler gear between the crankshaft gear and the gearbox input was prone to seizing on its dirt-filled ball bearings. The car should have had a Hy-Vo chain instead of that gear train, but it would have been too expensive for a very low-priced car made in a very low-productivity, fairly-high-labour-cost country.
