Use the power steering fluid that you get at your local parts store.
If you want to thoroughly flush the system and change fluid, removing the return hose and dropping it down to a pan is the easiest way. You described it as if it were some huge project. It should take no more than 10 or 15 minutes to remove the hose, drop it down into a drain pan, have someone start the car, pour two quarts throug the system while they turn the wheel, tell them to stop, reconnect the hose, fill the system while the helper is turning the wheel to bleed the system and put the reservoir top back on.
This is no big deal and is absolutely the best way to clean the system. There is no bleed screw because you don't need it. If you will read the manual you will see that bleeding is simply a matter of turning the wheel back and forth lock to lock a few times, then topping off the fluid level.
This is even the method that is the correct answer to a question on the ASE test. I can't think of any way that is any more thorough and is quicker and easier.
Good luck with it,