Bolditalic you can insist how much you want, doesn't change that this is an assumption. Your interpretation of RAI. You can homebrew this way, but RAW says the character gets the proficiency.
Multiclassing rules says that you get the proficiencies in the table instead of the initial proficiencies of the class, but you still gain features as normal, and clerics receive heavy armor proficiency through a feature. There's nothing saying about "proficiencies at first level", you are just making up this part.
If the rules were as you say, multiclassing rogues won't get expertise because it is a 1st-level feature, which is one of the main features of the exploration and interaction pillars of the class. Just like the bard too. And the proficiency Nature clerics gain at 1st level. Just imagine a war cleric without martial weapons wtf. Doesn't even make sense. Just like Ezekiel said earlier.