"Like it or not the Vatican City is regarded as a state. Therefore, it should be treated as a state visit UNTIL the time when it is no longer recognised as such internationally."
But the Pope and his ambassadors do not represent Vatican City State, they represent the Holy See and carry Diplomatic Passports from the same entity.
Likewise in return it is the Holy See with which states have diplomatic relations and not Vatican City.
Vatican City State and the Holy See are not the same thing.
The Holy See does not meet the Criteria under International Law for being a State.
However as said it has diplomatic relations with nations, which you can only have with a recognised state.
So one part of International Law makes perfectly clear that the Holy See is not a State and that therefore the Pope cannot be a Head of State. However another part of International Law says that as the Pope has diplomatic relations and diplomats he must be a head of state.
Mussolini saw the problems with this and that is why he created Vatican City. However the Catholic Church has always kept that separate from the Holy See, perhaps because it has lost all it's land before and doesn't wish to be tied to a physical place on earth.
So, as said, it's complicated and far from obvious or clear cut.
And I'm not actually stating my opinion on this question, just flagging up what the question is and why people are uncomfortable with the whole thing.