I went to the Auto-RX website, and noticed that it's possible to put
Auto-rx in the transmission too.  And from the instructions, it seems
critical to flush the auto-rx out after it's finished treating.

http://auto-rx.net/pages/applications.html

(begin quote)
For Do-It-Yourself Mechanics Only (technical expertise required)

If you are willing to do it yourself here's our recommendation. Install
6 oz's of Auto-RX and run 1000-1500 miles to clean the transmission.

THEN drop pan, change filter, reinstall. Measuring the old fluid in pan
into a milk bottle type container, install that amount of new ATF back
into transmission. Disconnect BOTH ATF cooler lines at radiator and
place a container or pan under each line end (helps to have an
assistant). Start engine, shut down after 1 or 2 quarts have come out
one of the lines. Refill exact amount of fresh fluid in transmission
(this keeps air out of the system).

Start engine again and repeat until clean fluid is coming out of cooler
line. This is messy and time consuming but will purge all old ATF from
transmission and torque converter. Reinstall cooler lines and start
car, cycle through gears, place in park and check level per owners
manual. Check for leaks. Top up as necessary after driving and heated
up. Don't overfill.

You just safely flushed the whole system, no T-TEC $ cost and Auto-RX
has removed the dirt and  contamination from your transmission.

(end quote)

A few comments though:

Wouldn't this leave some coolant inside the radiator?

To remove the coolant from the radiator, it makes sense to disconnect
only ONE ATF cooler line from the radiator.  One end should be the
suction, and the other should be the drain.

Let's say you disconnect the ATF cooler line on the RIGHT side.  (Right
side from looking at the car from under the hood.)  The metal tube
leads to the tranny, and the rubber hose leads to the radiator.  Would
the coolant be pumped out from the metal tube, or from the rubber hose?

Would it be safe to simply run the car, put a gallon of ATF on the
suction end, and collect the old ATF until it runs clear into a basin?