I had the exact same problem on my GDO. I found the following thread which suggested replacement of a capacitor. I replaced it and it's worked great since. Easy fix and costs very little. If you can solder/desolder, it's a trivial fix.
https://www.houzz.com/discussions/2381746/garage-door-remote-opens-but-wont-shut-door
I just read through that entire thread. Thanks for posting the link.
It appears my problem is not the same as in the thread. The problem in the thread is that the remote controls would open but would not subsequently close the garage door. The wall mounted button would work every time with just one touch--door opening or door closing.
There were two reasons in the thread you posted for this happening.
1. The user had installed an LED light bulb in the GDO. Once the door is opened via the remote (as when someone comes home in their vehicle) the light comes on and stays on for about 5 minutes, then automatically shuts off. The electronics in the LED light bulb base interfere with the radio signal from the remote telling the GDO to close. As long as the LED light is on, the remote control signal telling the GDO to close the door is blocked. Once the light turns off the remote control will work to close the garage door. Some posters said that they could get the remotes to lower the garage door when the lights were on by holding the remote control within a foot or so of the GDO main unit.
2. The capacitor is part of the timing circuit that keeps the GDO light on for 5 minutes until it automatically shuts off the light. The capacitor is old and no longer in specs. Something about the capacitor being old and defective is not allowing the signal from the remotes to be received by the GDO main unit. One poster in the thread said that he had incandescent lights and the remotes would work only after the timing circuit turned off the lights. Once the lights were off the remotes would work fine. This poster replaced the capacitor and now the remotes work regardless of the lights being on or off.
While I am handy with a soldering gun, in fact, have replaced swollen caps on circuit boards many times, I don't think the capacitor is the issue because my wall control unit will not lower the door unless I hold the button down. Everyone in the thread you posted said the wall control unit worked as normal, only the remote (wireless) controls had problems closing the garage door.
Just for fun, I went out and unscrewed both light bulbs and tried closing the garage door. It didn't work. I still had to hold down the button on the wall unit to close the door. I suppose I COULD try replacing the capacitor to see if makes a difference, but a.) I don't have much fun working in 20 degree weather, standing on a step ladder and craning my neck upwards in order to take out the circuit board to get at the capacitor, and b.) I've already made the decision to get a new door and GDO.
My Hail Mary plan: I've asked the buyer on Ebay if he will swap out the sending eye sensor he sent me for another identical unit he has listed. He has two more units listed on Ebay.