Problem is resolved
After many tries, I did get the visor Homelink working. Here's the trick:
The Genie GDO with rolling code only accepts 7 remote devices. In my case, I might have filled that quota already. I initially thought I had done everything right, several times, and it appeared the unit had a defect.
It is possible to clear the GDO memory by pressing and holding the learning button till the red light goes out. Then you press 'learn' once, LED blinks; press your hand-held unit once, light goes steady; press your handheld again, LED goes off. Repeat whole process for other handheld remotes.
Then, I cleared the visor unit by pressing and holding the two outside buttons (I and III) at the same time for about 20 seconds, until the amber visor LED starts rapid blinking.
Then you take your handheld remote, put it right close to the visor, and press and hold both its button and the button you wish to use on the visor until the visor LED rapid blinks. When you release, the LED turns green.
Now you press the 'learn' button on the GDO once, its LED blinks; you immediately press and hold the visor button you chose for about 2 seconds, until its LED changes from blinking to steady on; if you look at the GDO LED, it will have changed to steady on; then press and hold the visor button again for about 2 seconds until the LED changes from blinking to steady on. Look at the GDO LED--it should now be off.
You're done. Press and hold the visor button, and your GDO should operate. I noticed that you must hold the visor button pressed longer than my Genie handheld remotes--maybe a second or two--for it to work.
This is how to do the process with a Genie Excellerator GDO (a great GDO, I recommend it highly). I imagine that other rolling code GDOs are similar.
I hope this will help some other owners work this out on their own.