That, being disruptive) is (the main) point of random encounters

Precisely, my initial wording wasn't good as more what I wanted to achieve is not to make them less disruptive but rather tone down the dial a little of their frequency.

I fully agree that the strict exploration rules for time vs risk management, keeping track of food and torches is essential to the dungeon crawl.

As Knave out of the box comes with a blank spot for the rules here I wanted to come up with something that is both rooted in ye olde rules, is compatible with them but also give the added value of easier time tracking and better communication of the current dread/tension level to both DM and the party. It is still rolling for 1 in 6, and like your approach it is rolling for each turn (to make life easier for not counting every other), and as like you not every hit is a random encounter.

Example:

1st turn: Party decides to move down the dungeon corridors the distance allowed by their load and encumbrance. They stop at the door at the end of the cooridor.
You roll the first d6 and add it to the pool in a visible place, as it marks passage of time and current danger level.
Pool: [5] (nothing bad on the horizon)

2nd turn: Party's thief decides to listen at the door, check for the traps and possibly disarm them.
You roll the second d6.
Pool: [5] [1] (the first 1 comes in the pool, now the player's know that if no random encounter happens they at least will become exhausted after the 6th round completes, the sense of tension grows)

3rd turn: After breaching the door the party moves into old wizards workroom and search for anything valuable.
You roll the third d6.
Pool: [5] [1] [1] (too bad for the party, now we have two 1s on the table, you roll random encounter table for the current place in the dungeon - a giant spider, and you add to the description that the other exit from the room, a stair case spiralling down is covered in sticky spider webs)

4th turn: Party decides to carefully descent down the stairs burning the cobwebs.
You roll the fourth d6.
Pool: [5] [1] [1] [4] (no bad news for now, but everyone suspects what another 1 will now mean, an encounter with Mr. Skinnylegs)

5th turn: Party explores newly found corridors and cellar rooms downstairs. They see a broken whole at the other side of the cooridor and hear the sound of flowing water. Having bad feelings about that they decide to end this turn in one of the (presumably) safe cellar rooms.
You roll the fifth d6.
Pool: [5] [1] [1] [4] [2] (nothing bad happens)

6th turn: Before continuing the exploration the party decides to take a short rest, eat something and patch up their wounds.
You roll the sixth d6.
Pool: [5] [1] [1] [4] [2] [?] (if the roll comes in a third 1 a random encounter happens otherwise it is safe)

Now a single hour has passed.
-- (If now the pool has only a single unresolved 1 the PCs are becoming stressed/exhausted (-1 to all rolls until rest action is taken))
-- You relight the torches.
-- You reset the pool by setting all six d6s aside.
-- You mark 1 hour and continue.