It's ridiculously complicated, but currently JAL only releases F awards for travel on a particular day for 24 hours. When that window closes for that travel date, it remains closed for a few weeks.

On Thursday, July 23th 2020, F awards were only available for travel on Friday, June 18th, Friday, May 21st, Friday, April 23rd for Spring/Summer 2021
On Friday, July 24, 2020, F awards were only available for travel on Saturday, June 19th, Saturday, May 22nd, Saturday, April 24th for Spring/Summer 2021
On Saturday, July 25, 2020, F awards were only available for travel on Sunday, June 20th, Sunday, May 23rd, Sunday, April 25th for Spring/Summer 2021

So you can see how the award travel date window shifts over by a day every 24 hours. Today is Tuesday, September 15th which means that you can only book an F award for Wednesday, June 16th and Wednesday, July 28th at the moment.
You can see in the image below from JAL's website today that Wednesday, July 28th is the only date available to book a F award in July with AA because it is not wait listed. But if you check tomorrow you'll see that all F awards for travel on July 28th will be "Wait listed", but that booking F for July 29th is available (assuming someone didn't already snag the seats of course)





Meanwhile AA says JAL F awards are available for all dates, but will error out if you actually try to book it. But if you happen to attempt booking a JAL F award for a travel day where the award windows is "open", then you can book a JAL award flight on the AA website without any issues. You can check to see what days have F availability by using the JAL website. So booking JAL F with the AA website isn't broken at all *if there is an award actually available from JAL*, but unfortunately AA displays a whole bunch of flights where a JAL F award is not actually available, which makes this so confusing and frustrating.