The system was never tight. Look at the rest ruling in B/X. All you know is it is 1/5 ten minute turns of movement. Nothing is said about how characters rest, or if it can be interrupted.
@Ovinomancer has addressed some of this.
Its difficult to imagine what is happening here except that maybe you're mapping some kind of AD&D or 3.x or 5e "dungeon as ecology" paradigm onto this vs "dungeon as obstacle course" (which is basically what Moldvay is)? I mean, I don't even know why someone who has played Moldvay Basic would ask "can Rest be interrupted?" Since 1984 I've probably run 600+ dungeons (I never played a module...I've only run my own dungeons) and this has never_ever_ever (did I mention ever) been a thing...and its not clear to me how it would be.
Its simple. You've got 3 clocks you're monitoring, which I used (and still do) dice counters out on the table to represent these:
* Wandering Monsters - check every 2 turns; token or coin
* Rest - 1/6 turns and restart clock (players don't have to rest after turn 5...they can do it after 3...or 4); d6
* Light - 6 turns with torch (d6) or 24 w/ lantern (4d6) or 7-9 with the stupidly rare Light spell because its a 1st level spell so awful RoI and sunk cost (10)
Lets pretend its Turn 6 in the dungeon and the players performed their Rest 3 rounds ago in Turn 3. Order of operations:
* If there is a Stock encounter in this room, we handle the Encounter procedures. If not, the players perform their Exploration Turns. The torchbearer for the group is invariably going to light a new torch because theirs is going out at the beginning of next turn. We'll renew the Torch die at the beginning of the next turn.
* At the end of the turn, I check for Wandering Monsters. Let's say the result is no WM.
* Turn 7. No Wandering Monsters. No Stock Monsters. Move the Torch die and Rest die forward 1. Flip the WM token. The players at this point very likely decide to reset their Rest die because they don't have an Encounter (so no possible -1 to hit/damage for another 6 turns).
Rinse and repeat. The players are monitoring WM, their light, their rest and managing their resources/load-outs/exploration while they do this. And they're managing their treasure accrued in accordance with their Encumbrance, protecting their Porter(s) to extend this, and eschewing Coin for treasure that has a better $:Weight ratio.
Why does 5e not do this loop as well as Moldvay Basic?
* Light is a trivial resource in 5e. Not only is the equipment portion lest cost for significantly greater return, but Light is a Cantrip rather than a 1st level spell (and spells are much more proliferate and Rituals exacerbate this). Light management is a big ole nothingburger in 5e. In Moldvay, its a deal in multiple ways. Basically any spellcaster obviates this concern.
* Encumbrance is a relative nothingburger in 5e compared to Moldvay and the handling of it is made worse because Moldvay's is trivially "Coin Weight." So my guess is most people trying to run 5e as a Dungeon Crawler don't even pay attention to this.
* 5e PCs are massively more capable in Exploration than their counterparts in Moldvay.
* 5e PCs are massively more capable in Combat than their counterparts in Moldvay.
* 5e PCs have access to more prolific social obviation spells/tools that their counterparts in Moldvay.
* HP restoration for 5e PCs is profoundly easier than their counterparts in Moldvay.
* Wandering Monsters + Rest don't exist in 5e. You're going to have to bolt that on and then try to integrate it with all of the rest of the system (duration timers for spells and what not). I mean, if you artificially tacked on the Exhaustion Rules w/o a 10 minute Rest every hour of exploration and then hacked on some Wandering Monsters...would this really be that much of a pressure point for 5e characters? Particularly with the fact that all of the other stuff aren't actual pressure points in this edition? I don't see it. And again, even if someone did get Exhaustion, you can offset that with Help/Advantage or Inspiration or Bard/Cleric. The means that 5e characters have to marshal aren't just a cut above...they're about 5 cuts above at both the individual level and ESPECIALLY at the Team PC synergy level.
Add that all up? Ze game does not remain the same (I think the saying was). In every way possible, the system promotes profoundly more competency at the lowest levels of play and reduces the pressure points to nothingburgers while simultaneously not including the key, integrated Exploration tech of Moldvay Basic.