I'm not saying Deere is bad, but I have a hard time believing it is THE best. Everyone is getting base stock and additive packages from the same handful of sources, especially if you are talking about synthetics. For PAO synthetics I've been told there are only a couple suppliers of base stock in the North America. And when you get down to it everything is a compromise - antiwear, viscosity, VI improvers, anti-oxidants, water control, high pressure additives, compatibility with clutch/brake pads, compatibility with copper, etc. What it usually comes down to is what is the best combination for the particular machine and application.
For what its worth IH pioneered "universal tractor oil". Deere's Hy-Gard is a copy of IH Hy-Tran. I have no hard data to say which, if either, is better at this point in time so please don't start a war over which oil is best.

These oils have a lot of additives - high pressure for hypoid gears, water absorbing to deal with long storage periods, wet clutch and brake compatibility. Typical hydraulic fluid has none of these and is hydrophobic, allowing water to settle out.
To add to BrokenTrack's comments power steering fluid and ATF will both have a fairly high viscosity index, which means less change in viscosity vs. temperature than hydraulic or universal tractor oil. ATF has a lot of anti-foam, anti-oxidation to deal with the heat and friction pad additives which you don't need. And you didn't mention if the pump is used, but I'm going to assume it is. In this case I'd stick with power steering fluid. It is cheaper than ATF and you don't have to worry about seal compatibility in the pump. Making a major change to the additive package could cause the shaft seal to shrink or swell.
ISZ