Setting a time limit up-front is one way to start an adventure, but by no means the only one, and one which I wouldn't want to repeat regularly.
You're not expected to. Keep time pressure (either via timed quests or hostile environments/ 'random' monsters) on the party for around 50 percent of the time.
They'll self regulate much of the rest, and the odd nova strike is actually pretty good to let them get away with.
There is nothing wrong with a single encounter adventuring day (of increased difficulty) if its used sparingly.
Still the question remains: how do you make your design work without limiting the players' freedom? The whole system an adventuring day, encounter budgets and different refresh-speeds of powers/abilities seems to be akin with 4e's system, which is good for some types of adventures and a problem for others.
The players are free to do what they want. You lay the hook out and they follow it; or they dont. See the quick off the cuff adventure I posted in this thread as an example.
This is the case in most timed adventures. All that happens if they dont do it by time X is they suffer a drawback of some kind. No different to everything in the real world.
This again is a meta-game reaction in order to teach players a desired behaviour fitting with the adventure and game system. This flexible, movie script-like adventure design just isn't my favourite way of handling things for most adventures.
Nothing is stopping you from deviating from this meta, but it drastically alters the balance of the classes and the difficulty of the encounters.
The whole thing just means that 5e might not be the best edition for me.
There are options in the 5E DMG to tinker with the rules to better fit your style of play.
For example there is the 'longer rest' variant where short rests are overnight affairs, and 'long rests' are week long numbers. Its perfect for campaigns that only feature 0-2 encounters per in game day (urban, pirate, wilderness and city adventures).
For your more typical dungeon crawl, the standard rest paradigm works best. Its unreal to think of a party in 1-4E walking into a dungeon, nuking the first room, then falling back to rest for 8 hours. I shudder at the players that thought that was OK to do, or the DM that let them get away with it.