The rules don't say that.
Certainly it does, which I have highlighted in the spell repeatedly yet you seem determined to ignore.
Here, one more time, maybe it will eventually sink in?
When you (the cleric) casts the spell, you (the cleric) make a melee spell attack.
And since you like this so much:
"Your attack (the cleric) bonus with the spell attack..." See that?
You (the cleric) are hitting the
fire shield target with the spell effect, because in order for the spell effect (YOUR spell effect) to hit, you must attack with it. The
spiritual weapon spell is what you are hitting with, when you succeed and determine "whether the spell effect hits".
The spell effect is nothing, literally, without the cleric making the attack and hitting with it.
Your argument for "natural language" doesn't prove your point of view when you are focused solely on the phrase "spell effect hits". It doesn't hit without you attacking with it, in which case YOU are hitting the target with the spell effect, just as a fighter or whoever hits their target with a weapon attack.