I'm still trying to figure out what a reliable or intended strategy for surviving 1st level is supposed to look like. In part so that I can design a dungeon that supports it, or at worst not design a dungeon that accidentally hard-counters it.
Assume for the sake of argument a Big Four party of fighter, MU, cleric, thief. Total XP required to get them all to 2nd is 7200. Assuming stats roughly typical for 3d6, with +5% prime reqs all around and not much else, bringing us down to about 6850 XP. Assume also that we're following Moldvay's advice to not let new players lean on hirelings as they learn the game.
A single dungeon encounter of 2d4 (average 5) goblins is a pretty fair match outside of the party's one sleep per day. 3d4 (average 7.5) skeletons are liable to get messy if the cleric fails the turning roll. And those are both bite-sized random encounters with pocket change at best for treasure. Lairs are much bigger, and dividing and conquering them requires shared languages, decent luck with reaction rolls, and ideally multiple humanoid lairs to pit against each other - but we can't count on getting that from the stocking tables.
Fighting monsters for their treasure seems like a pretty bad idea at 1st level. You don't have any healing resources and your one sleep might be best used to survive a fight that you didn't want to be in, rather than offensively to win a fight that probably won't have treasure. Oil and war-dogs are great and all, but... is that really the intended solution?
Is it possible that you're meant to spend first level skulking about, looking for trapped or unguarded treasure and avoiding almost all encounters?
Well... I'm not sure the math works out. In ACKS, the expected value of an unguarded treasure on the first level of a dungeon is 870 GP, so you'd need about 8 unguarded treasures to get everyone to 2nd, for that cure light and extra sleep per day (the thief, incidentally, will level after only about four such treasures). Buuut between trap rooms and empty rooms, about 9% of rooms (1 in 11) have treasure but no monsters. So on average you'd need to explore 88 rooms on the first level of the dungeon to level by unguarded+trapped treasure, and that's assuming no casualties to traps, random encounters, stumbling into lairs, etc (granted, also assuming no monster XP - but if you're avoiding lairs and start heading for the exit when you've burnt your one sleep on a random encounter, you're talking maybe 25-50 monster XP per expedition). 88 rooms is huge for the first level of a dungeon - when I'm building standalone non-mega dungeons, I often have about that many rooms across three levels. But it's sort of plausible that one could build a ruin that large, all of dungeon level 1. 88 rooms is also going to take quite a lot of play time to get through - my players in Rat-hell were exploring maybe 10 rooms per session, so we're looking at 9 sessions without any resets from casualties. That seems like a long time to spend at 1st level.
The situation is much worse in OSE. It's still 9% of rooms with treasure but no monsters, but the expected value of that treasure on the first level of the dungeon is a paltry 160gp - about a fifth of what it is in ACKS. So you'd need to explore over 400 first-level dungeon rooms to level everyone off of unguarded treasure in OSE.
Is the answer to go deeper? If you're already adopting a Robinsonian posture towards monsters, and the unguarded loot is better at lower levels, maybe this isn't crazy. And traps don't necessarily scale up when you go down; ACKS' list of traps is the same for 1st-3rd dungeon levels, and OSE doesn't specify. 2nd level unguarded treasure in OSE has an expected value of 492 gp (about triple), while in ACKS it's 1491 gp (almost double). And many of the random encounters on the 2nd level dungeon table are still susceptible to sleep - 5 2HD lizardmen vs 9 HD affected on average by sleep, for example. So a 2nd level random encounter isn't necessarily any more of a game-ender than a 1st-level one is...
With ACKS' treasure numbers, I could see a dungeon with 30 rooms in the first level and 30 rooms in the second level yielding enough unguarded/trapped treasure to get a party to 2nd. 'course, in those 60 rooms you've got about 20 monster rooms, of which probably 2 are lairs...
There is also the question of surviving the traps. In 60 rooms, we're talking about 20 traps (of which 6-7 have treasure). Most of the "treasure traps" are threats only to whoever interacts with the treasure, but some of the "room traps" are threats to the whole party at once. In 20 trapped rooms, we'll probably pull the "room fills with poison gas" one at least once, and it could easily happen twice. Saves at 1st are pretty bad (we should expect ~1.5 survivors based on saves). Finding and Removing Traps at 1st is also pretty terrible. The Trapfinding proficiency in ACKS is actually a huge boost at this point in the game, almost doubling the thief's chances of successfully finding or removing a trap (granted, it's 15% chance of success to 25%, which is still not great). ACKS also removes the poison gas room trap, though the falling bricks from the ceiling for 2d6 is arguably worse at this point since Petrification and Paralysis saves are slightly worse than Poison and Death, and its average damage is still going to be enough for the vast majority of 1st-level characters.
On the other hand, when you frame Finding Traps as "15% chance to detect non-monster TPKs before they fire", thieves start to sound pretty good. It's not a high chance, but it's better than nothing.