I believe a voluntary change triggers a reprice with the current fares so that passengers cannot skirt advance purchase requirements. This may be different if travel has already commenced.

AA has a quite generous flat tire rule that allows you to get a confirmed seat if you are at the check-in counter up to 15 minutes after your flight departs. At outstations, the issue is that the AA check-in counter might be closed by the time you get there. I would still keep it in mind if your LA flight is delayed.

Same-day flight change (and standby) on AA is only applicable to domestic flights and JFK-LHR, as you note.