I enabled the Troop Setting “Let scouts edit their own achievement records”. When my son goes to Achieve->“Record Individual Progress”, it will save the Notes but not the dates that he enters. It doesn’t matter whether he types in the date or uses the calendar widget.
I just did a bunch of testing and cannot reproduce the problem you are describing. One thing I did find is the Achieve → Record Individual Progress is actually tied to the TT Setting, “Allow members to enter provisional progress (to be approved by a leader)”, if that was off then I got a permission error attempting to load the page address and the Record Individual Progress does not show up in the Achieve menu. When the “Let scouts edit their own achievement records” then I could just go to the Achievements tab of my youth user Profile and enter achievements directly there. It seems to me the date field issue may not actually have to do with the setting but may just be a browser issue. Have your son try a different date field under his user profile, Achievements and enter something there. If the date field still isn’t working then I would suggest clear browser cache and reset browser to see if that clears up the issue. You could also log in to your son’s account on your computer and see if you have the same issue.