I finally got around to calling the Wilderness Permit number and my regular Inyo NF staff person-contact. Good changes are ahead!
Software is currently in development -- not in testing yet, but hoping to be in production by July. The idea is to automatically make walk-in permits available 2 weeks before the hike date at 7 AM each morning. Also part of that is to offer refunds to people who cancel their reservations up to 12 days before their hike date. (I am lobbying that this refund window be made way shorter -- I don't know what bad could happen if they made it as short as several days ...except Inyo needs to make contact with the recipient to email their permit.)
Unfortunately Whitney lottery permits are not being offered that same refund for cancellation. Apparently lottery permits are handled as a different animal. I pointed out that cancellation refunds would make a large number of permits available for others to grab (and pay for) so Inyo and recreation.gov would actually be rewarded for the policy.
Also, there is some mechanism for Whitney lottery no-show permits to automatically go back online, so there may be more last-minute permits available.
As for true "walk-in" permits: Due to COVID, the Visitor Center and face-to-face encounters with staff is still being limited or non-existent. Change to get back to pre-COVID happens slowly.
I do think one of the biggest issues that causes some of the longer time windows is the need to get an official hard-copy permit into the hiker's hands that they can carry.
By the way, everyone, there are a really good number of ideas presented in this discussion. Occasionally, Inyo staff read these discussions, so your time to write is appreciated. The reservation system is in constant change due the the changing environment of COVID as well as the Internet and online environment. I think the process will continue to improve over time.