Short rests are not enough. The group I ran for took two. Clearly, the group can choose not to partake in the fifth, sixth, or seventh mission if they feel like they can't accomplish it. ("Sorry Governor, we understand that the townsfolk in the Church of Chauntea are about to be murdered by kobolds and cultists but we just can't help you anymore.")
I totally disagree. It is strictly a problem with the adventure itself if you only have 7 hours to complete critical missions and CANNOT take a long rest in order to face those challenges.
When I play tested this adventure for my veteran group of players at Gencon it was clear to me that they would not be able to survive another session and complete the remaining missions if I did not change the adventure, omit certain missions,
I had a much longer reply but it boils down to this. You are mixing up the fact that there are 7
provided missions by thinking there are 7
critical missions. I'm sorry but it seems you are ignoring or misread the part where
you are not required to put your players through all missions. Just because they're there doesn't mean they're mandatory, required, or critical. AT ALL.
If the party took 2 short rests then they would have 2 less missions to take on, 5 total and not 7, remove 2 from the remain roster of potential events. If they took a 3rd rest that would be 4 total and not 7, choose which one is left. You are expected to allow this, you're not 'correcting' the adventure by doing this. You're expected to run the number of missions as appropriate to your table. Saying "sorry, that's what it says in the adventure" is not valid. Especially when the text explicitly assigns the responsibility to you, the DM, and instructs you on how to treat the number of mission and rests within that timeframe.
From a story-telling stand point, the characters don't know what carnage is happening out there, and it hasn't happened yet. They don't know what they're missing by resting (until you tell them). It's up to you to 'write' your fiction and decide which (if any) of the missions the party missed actually happened. Either way is legitimate.
... or supply them with help that the adventure does not offer. Since we were play testing, I offered this information to my players who decided that they would like to see just how tough it would be "as written." They TPK'd in the second encounter of the second session (it was part one of the Sally Gate Mission).
If you presented your players with their predicament and they chose to 'cowboy up' then that's an out-of-character decision by
players wanting not to miss any part of the adventure options and your decision to run them through it. Which is a perfectly legitimate exercise, but it's not actually the adventure "as written".
I think if reading the adventure that wasn't clear in the first place,
it would be useful to listen to Kobold's intention behind the adventure design (specifically from min 26 where they address exactly this episode and this question).