Both Forge and the Fabric Api use an accessWidener to make ModelPredicateProviderRegistry.register(Identifier id, UnclampedModelPredicateProvider provider) and ModelPredicateProviderRegistry.register(Item item, Identifier id, UnclampedModelPredicateProvider provider) public.
They should be be made public in the common sub project too so uncommon code isn't required in a mod to access them.

No, forge modifies ItemProperties.registerGeneric's method signature, making them not binary compatible, use architectury API's ItemPropertiesRegistry abtraction.