If the rule books for 5e just dropped out of the sky there are a few sources that would point players and DMs to using initiative to track things other than combat. The first of these is on pg 5 of the PHB:
Unlike a game of make-believe, D&D gives structure to the stories, a way of determining the consequences of the adventurers' action. Players roll dice to resolve whether their attacks hit or miss or whether their adventurers can scale a cliff, roll away from the strike of a magical lightning bolt, or pull off some other dangerous task. Anything is possible, but the dice make some outcomes more probable than others.
Dungeon Master (DM): OK, one at a time. Phillip, you're looking at the gargoyles?
Phillip: Yeah. Is there any hint they might be creatures and not decorations?
DM: Make an Intelligence check.
Phillip: Does my Investigation skill apply?
Phillip (rolling a d20): Ugh. Seven.
DM: They look like decorations to you. And Amy, Riva is checking out the drawbridge?
The next is on pg 7 of the PHB:
- The DM narrates the results of the adventurers' actions.
In certain situations, particularly combat, the action is more structured and the players (and DM) do take turns choosing and resolving actions. But most of the time, play is fluid and flexible, adapting to the circumstances of the adventure.
Combining this with a portion of the note about Combat in the PHB on pg 8:
Combat is the most structured element of a D&D session, with creatures taking turns to make sure that everyone gets a chance to act.
along with the text describing initiative on pg 177 of the PHB:
At the beginning of every combat, you roll initiative by making a Dexterity check. Initiative determines the order of creatures' turns in combat, as described in chapter 9.
and the text you referenced on pg 181 of the PHB:
In situations where keeping track of the passage of time is important, the DM determines the time a task requires. The DM might use a different time scale depending on the context of the situation at hand.
In combat and other fast-paced situations, the game relies on rounds, a 6-second span of time described in chapter 9.
Gives us a good idea that non-combat fast-paced situations should use the rounds system defined in the combat chapter (PHB pg 189):
The game organizes the chaos of combat into a cycle of rounds and turns. A
round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other.
Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.
The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.
The DMG then gives two concrete example of using initiative in non-combat situations Chases (DMG pg 252):
BEGINNING A CHASE
A chase requires a quarry and at least one pursuer. Any participants not already in initiative order must roll initiative.
and Complex Traps (DMG pg 121)
Complex traps work like standard traps, except once activated they execute a series of actions each round.
When a complex trap activates, it rolls initiative.
It also has a good example of an effect that suggests close tracking of time is appropriate, Planar Portals on pg 45 of the DMG:
Time. The portal functions only at particular times: during a full moon on the Material Plane, or every ten days, or when the stars are in a particular position. Once it opens, such a portal remains open for a limited time, such as for three days following the full moon, or for an hour, or for ld4 + 1 rounds.
Bringing all of that information together we would be able to know that we can come up with inventive ways to use the initiative and rounds system to track non-combat situations.
Other Initiative Possibilities
The DMG also gives the option of Side Initiative on pg 270, which might fit what you are looking for, of close tracking of time, while also giving the players order flexibility.
Under this variant, the players roll a d20 for their initiative as a group, or side. You also roll a d20. Neither roll receives any modifiers. Whoever rolls highest wins initiative. In case of a tie, keep rerolling until the tie is broken.
When it's a side's turn, the members of that side can act in any order they choose. Once everyone on the side has taken a turn, the other side goes. Once everyone on the side has taken a turn, the other side goes. A round ends when both sides have completed their turns.
I've put together an example of applying this initiative variant to a non-combat situation. Drawing on your option of a non-dangerous round-based effect, I think Planar Portals with a time based requirement (detailed above) fits the bill.
If we take the round based option a brief idea for using round based initiative for interacting with the portal:
Party is split searching a large room after beating a BBEG. One of the party leans on the wall for support while investigating, accidentally pushes a recessed stone tile, which starts glowing.
DM: Roll Initiative as a group, you don't see any threats and as a result you are not in combat, but I need to track time pretty closely for a little bit
Party rolls 15, Portal rolls 12 (DM Knowledge: Portal appears in 2 rounds, lasts for 6 rounds, Stone Button glows for 8 rounds, glow increasing until round 4, after which it starts to fade)
DM: What do you guys do? (to the party)
PC who pressed the button: I shout at the others "There is a stone that started glowing over here people!!"
Mage PC: I run over towards the glowing stone. Do I get there?
DM: Yes, do you want to do something while you were there with your actions?
Mage PC: I would like to see if I know or can discern what it is
DM: Roll an Arcana check
After everyone in the party has decided to do/not do something the DM describes what happens with the Stone/Portal
After the event has finished:
DM: Ok we are no longer in initiative any more