We tend to use Backgrounds in our OSE and B/X games. I even have a table that Players can Roll/Choose from here: Character Backgrounds & Prior Vocations.
To me, the purpose of a Background is far more compelling when it isn’t boiled down to a series of Mechanical Bonuses or little Numerical Details. The only change we make is there’s another line on the Character Sheet where you can fill in your “Prior Vocation.”
Virtually all “sub-classes” fit very comfortably within the archetypes provided by the Core Classes provided by B/X without anything other than maybe a slight bit of re-skinning in the fiction. I mainly just ask myself "Does this Class solve problems through direct application of physical effort? (Fighter!) or are they more into indirect applications (Thief) Do they rely on Magic to directly address Challenges? (Magic-User) Or is there Magic more indirect in nature? (Cleric!) I try to view the Classes as less vocational and more as just a suite of Tools that are provided to Characters to address some of the more common challenges found in these games. You can write whatever you want in that box, but “under the hood” we’ll just use the same XP Progressions, HD, Weapon/Armor Restrictions, Saving Throws as one of the conventional Core Classes.
However, filling in that Background can represent a wealth of information though: It’s a multitude of Skills, Talents, Specialized Knowledges, Unusual Knacks, and even sometimes more physical features. Rather than having to reference a big list, or have the Players actions guided more by what they can and can’t accomplish mechanically based on the Skills listed on their Sheet, we just have a single line for Background to encompass all this stuff.
It’s very simple: The Background just adds another layer of Competency for related tasks. If prior to becoming an Adventurer, you were a Sailor then I’m not going to make you roll the dice to tie a complex knot, swim in normal circumstances, locate North by the Stars, scale a fish, or mend a net. If you get in a grog drinking or sea shanty contest with others, you will have better chances at succeeding than someone without this background, etc.
I tend to Play with pretty mature people, so other the occasional humorous joke, I don’t see a lot of attempts to exploit this system. We do acknowledge that sometimes these Backgrounds might be able to be leveraged in Unusual or Interesting ways though. A Character who is a former Blacksmith can easily make a Horseshoe, keep a Fire lit for a long time, etc. but when it comes to grabbing a white-hot Metal Idol from a Dias…they might end up suffering less harm than others due to their Calloused Hands by invoking that past history of working with very hot objects.
So when adopting this principle to a B/X style game, I’d really lean more on the Equipment Packages option of the ones you’ve listed. Gear can be a very powerful Problem Solver in these games. B/X doesn’t really make a lot of provisions for highly specialized Skills outside of the normal Dungeoneering/Exploration x-in-6 Checks (Stealth = Surprise, Listen = Hear Noise, Search, Hunting/Foraging, Open Doors…and the oft overlooked Tinderbox Check.) I don’t really consider the Special Class Abilities of the Thief Class to be quite the same, as those abilities are more to define their niche within the context of Teamwork (like a Magic User’s Spells, only with less limitations in terms of Resources.)