Sub-question: What about Pokemon that can change Form(e)s? Can Shaymin learn Rock moves because it's Sky Forme is weak to Rock? Does Rotom get ALL the moves? If yes, does this apply to in-battle Form(e) changes too, such as Meloetta gaining access to Flying, Psychic, and Fairy moves due to its Pirouette Forme being weak to those? (Would Darmanitan
have to be running Zen Mode as its Ability to get Dark and Ghost moves, if so?)
Also, is this only released moves? The OP doesn't say either way. (Really, the OP could stand to be improved in general)
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The rest of the types.
Steel weakness is an interesting grab bag of options, but the four big things (Aside from coverage) are: King's Shield, Shift Gear, Metal Burst, and Bullet Punch providing priority. An obvious point off the bat is that most Rock types are weak to Steel and slow and bulky Physical attackers, and so will love Shift Gear. Several Fairies have much worse Defense than Special Defense, and will appreciate King's Shield.
Fire weakness of course provides incredible offensive potential, but Will O Wisp, Sacred Fire, and Mystical Fire are somewhat more interesting with the latter potentially allowing a Physical wall of some sort -say, one of the many Steel types- to hamper attempted Special attacker switch-ins, while spreading Burns is just good stuff all around.
Water weakness is mostly notable for providing priority -too bad no Skill Link Pokemon is weak to Water, otherwise Water Shuriken would be awesome- and providing an excellent option for spreading Burns. (Which isn't very helpful to the Fire types who are included in this number) The coverage is the primary other benefit, in practice.
Electric weakness provides a grab bag of weird utility moves that mostly aren't very useful, alas, but it includes the ability to throw around Paralysis and it's certainly worth commentary that Mega Pidgeot gets Zap Cannon!
Fighting weakness provides: a fixed damage move, the only Special priority in the game, Physical priority, damaging forced-switching, a suicide move that only cares about HP of the user and target, Counter, partial Flying coverage (Flying Press), Bulk Up, Sacred Sword and Storm Throw for bypassing defensive boosting, a leeching move, and of course tons and tons of raw firepower. It's good stuff.
Dark weakness provides the invaluable Knock Off, Nasty Plot as a poor man's Tail Glow, Dark Void, Foul Play (Which is excellent, given how Psychic and Ghost types generally can't pull off a decent Physical move but may appreciate it anyway), Hone Claws for those powerful but inaccurate Physical attacks, Parting Shot (Can even be paired with Prankster and Regenerator!), Topsy-Turvy (Again, Prankster), the strongest priority that can be passed out by Super Effectivemons, Taunt, Switcheroo... oh, and there's the actual coverage component too.
Poison weakness gives Fairies and Grass types relatively widespread access to Toxic Spikes, Gastro Acid (So for instance Clefable can suppress Mega Gyarados' Mold Breaker, denying it the ability to boost past it), Coil (Now if only more of the two types are Physically oriented...), Acid Spray (An excellent tool for ruining Clefable, for one) and of course coverage.
Ice weakness provides priority, Haze (Prankster Haze from Whimsicott can be used as a form of anti-setup), Glaciate should always be taken over icy Wind (Well, not for Technician Roserade or Technician Breloom), Freeze Dry, Frost Breath to push past defensive boosting, Mist for protecting from stat reduction if you think it's worth the move slot/turn, and of course coverage.
Ghost weakness almost universally overlaps with Dark weakness in occurring, and as coverage runs into the flaw that you'd usually rather use Dark for coverage for a few different reasons. It does, however, slightly help a few Ghosts -Golurk can run Shadow Claw instead of Shadow Punch, for instance- and it does provide priority that doesn't require the enemy is attacking, Curse for Physical boosting if you can't get Bulk Up, a fixed damage move for stallmons, the obnoxious Destiny Bond, and the quirky Trick Or Treat -combine with Pursuit?
Ground weakness is pretty much entirely coverage, largely. And Spikes. Can't forget Spikes stacking.
Rock weakness of course spreads the Stealth Rock love, but it also provides Speed boosting. (Rock Polish) Mostly it's coverage, but the coverage is
hugely improved, with widespread Power Gem access and Diamond Storm access making reliable Rock coverage
much more of a thing, coming from Fire, Flying, Bug, and Ice types. Entei certainly appreciates replacing Stone Edge with Diamond Storm.
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Next time I'm liable to look at this from the opposite angle: the "default" intersections... such as how pure Fire is Edgequake all by itself.