I think it would be nice to have access to the shop before starting a day, to purchase any shop add-ons/check store stocks/look at achievements/view accumulated days (considering you can’t at the ‘Next Day In xx:xx’ screen). The ‘Next Day In xx:xx’ screen should be completely done away with. If there is some magic way to manage your shop while waiting already, disregard this post, but I haven’t been able to see any way to do it yet.
I think the button to start the next day should just appear at the bottom left corner of the shop screen (replacing the pause button when a day isn’t already in progress), and the timer should be placed just above that (replacing the accumulated days counter when it reaches zero to signify you have no more days).
Also, just to cover all of the bases, I feel that clicking continue on the post-day report screen should take you back to the shop with no day in progress (even if you still have accumulated days), to allow you to get anything you need done before the day starts if you feel like doing so, and allow you to click the ‘Start Day’ button when you decide to.