Don't Cut the Corners

Dungeon adventuring is survival horror!
The genius of White Box and the hobby in general is that the game can be played in so many ways. I have always been a huge fan of the dungeon. Darkness, limited resources, risk management, greed - all these play a significant role in the dungeon delve done right. Nothing scares the imaginary dickens out of a bunch of pen-and-paper would-be heroes like being plunged into darkness where nobody but the baddies can see anything.
"What's in front of me? What's behind me?  What's that noise?" When players imagine a situation where their characters are without light, panic is often the first response. As a referee, we need to use that. Demi-humans who can see in the dark really spoil this "tool" as well as that of surprise. Walking around the underground carrying a light source pretty much means the party is seen before seeing. Rolling for surprise is a regular thing. The party can really only surprise baddies by opening a door, otherwise the lurkers-in-the-dark see the party of adventurers coming by their light source.
Encumbrance may be cumbersome, but unless the referee makes some allowance for the concept resources like torches, food, holy water, oil, pitons, rope, arrows all become unlimited. Being down to one's last two arrows is an experience that heightens tension and verisimilitude. The referee should use all the advantages in their toolbox in an effort to provide the most immersive game possible. Otherwise we risk a "grab the loot and level-up" game that quickly loses its shine.
Weight carried is especially important. When loot is discovered in a large quantity and must be carried to the surface in order to earn those coveted experience points (XP) how much a character may carry is important. Vol. 1: Men & Magic gives some advice regarding encumbrance, but like the author says, this can be taken as guidelines or suggestions and I have seen encumbrance systems I like better and which seem more usable at the table (Lamentations of the Flame Princess just counts items). The most important thing is to have a system - almost any system is better than a "hand-wave".
Competition can also be used to heighten the tension and verisimilitude of the dungeon experience. If it is not possible to have at least two play-groups exploring the same dungeon (different players, different sessions) then the referee may want to devise a group of adventurers who are also known to frequent the dungeon and who may get to that treasure horde first. In addition to offering some excellent opportunities for role-play, the occasional encounter with other delvers (NPC or PCs) can expand potential plot-lines, introduce help or hindrance, and add novelty to a dungeon.
Playing "It's a game about leveling-up your character" can end up being the "more, better fun" lie. Gary Gygax frequently wrote about the danger of giving away too much too quickly and thereby ruining the game. He wasn't talking about plot clues! The dungeon is made up of "levels", some of which (lower levels) are at first too difficult or "dangerous" for the beginning characters. The lower levels are meant to "open up" as characters advance making the game continuously challenging and novel. Unlike the non-dungeon "adventure path" style of play where risk management can become the responsibility of the referee, the dungeon presents the players with an ability to manage risk to their PCs by choosing how deep to delve. Deciding on whether to head down those newly discovered stairs into a lower level and higher danger can be an opportunity for role-play among the players and entertainment for the referee.
Pacing of character level advancement can be somewhat of an individual preference, but one level per 4-6 sessions, perhaps doubling to 8-12 sessions per level at higher levels, seems to fit with impressions I get from reading some of the comments made online by Gary Gygax, Tim Kask, Rob Kuntz and others regarding level progression in the original game when they talk about it taking a couple of years of weekly play for a character to reach "name level". The "reward" in such play is the fun that happens at the table and in becoming a better player of the game (game mastery). By contrast, if reward is defined as leveling up, then "rewarding" players for showing up by leveling their characters becomes their fun. Cutting the corners misses the point.