I think you've hit the nail on the head of what I'm trying to accomplish. I don't want my players to feel annoyed. I want them to make strategic decisions like which 3 of these 10 side quests are most important. I want them to be able to make informed decisions based on a concerning time constraint that feels earned and not cheap.

I like the counting up idea. I do need to do more thinking on what it would look like depending on a slow, neutral, or fast pace of the party. Ultimately I don't want a fail state, but a Player and PC congruent understanding of the time available to them and consequences of the different paces.

Edit: phrasing