Mintz, sorry, but you seem to be the issue, really, not the reminder thing. It's only my opinion, so take it for what it's worth.
Anyway, for full press, you have to find the time regularly and you can't really say (nor think) that you need to be reminded to save your orders. Just because you need to log on often, it will be natural to check the site.
Your suggestion could be good speaking about no-press; in my opinion, just informing that a game started could be great.
But no, FD, I don't want to implement it :P